Quadratic and Linear Equations

Sum and product of roots

For a quadratic equation ax^2+bx+c=0 with a\neq 0, let the roots be \alpha and \beta. Then \alpha+\beta=-\dfrac{b}{a} and \alpha\beta=\dfrac{c}{a}. These are the relations of Vieta, and they let you read the sum and product of the roots directly from the coefficients without solving the equation. The most common trap is forgetting the leading coefficient: the sum is -\dfrac{b}{a}, not -b, so a must be divided out even when it is not 1. For example, the sum of the roots of 3x^2+5x-2=0 is -\dfrac{5}{3} and the product is -\dfrac{2}{3}.

Conversely, the sum and product fix the quadratic. The monic equation x^2-(\alpha+\beta)x+\alpha\beta=0 has the given roots, and any other equation with the same roots is this one multiplied by a nonzero constant. So when the sum and product are specified, exactly one monic quadratic exists.

A single condition relating the sum and product gives only one relation among the coefficients, and this leaves freedom. In ax^2+bx+c=0, the requirement that the sum equal the product gives -\dfrac{b}{a}=\dfrac{c}{a}, so c=-b while a\neq 0 is free. Hence infinitely many quadratics satisfy the condition.

Symmetric expressions in the roots are written in terms of \alpha+\beta and \alpha\beta. The sum of squares is \alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta, and when this is set equal to a given value, the resulting linear equation in the product gives the unknown constant term. Higher powers follow the same pattern: \alpha^4+\beta^4=(\alpha^2+\beta^2)^2-2(\alpha\beta)^2. Do not confuse the sum of squares with the square of the sum. If the sum of the roots is 5 and the product is 2, the sum of squares is 25-4=21, not 25.

The difference of the roots is found from (\alpha-\beta)^2=(\alpha+\beta)^2-4\alpha\beta, and taking the square root gives |\alpha-\beta|. Squaring removes the sign, so a parameter that arises from squaring must be examined for both signs.

If a root is known, substitute it into the equation to find an unknown coefficient, and then use the sum and product to find the other quantities. Once the coefficient is determined, the requested sum of squares is computed from the full equation. When the unknown appears inside the coefficients themselves, the sum and product give a system of equations in the parameter, and both relations must be used together. Dividing by an expression is valid only when that expression is nonzero, which is why a stated nonzero condition is important.

If the sum of the roots is zero, the coefficient of x must vanish, so the equation reduces to ax^2+c=0. Its roots are \pm r, and their difference is 2r.

To form the equation whose roots are modified versions of given roots, first find the original sum and product, apply the transformation to each root, compute the new sum and product, and rebuild the quadratic. Old coefficients should not be reused directly. When two numbers are determined by their sum and product, they are the roots of t^2-(\text{sum})t+(\text{product})=0. For a sum of 9 and a product of 14, the numbers are 2 and 7. Factor pairs help when the data are integers.

For a rational equation, clearing denominators produces a quadratic, and the roots of the new equation are often the squares of the old roots. The sum of squares and the product of squares follow from \alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta and \alpha^2\beta^2=(\alpha\beta)^2.

For a quadratic with real coefficients, one known root determines the other through either the sum relation or the product relation. Real coefficients do not force the other root to be the conjugate of a surd root, since that rule applies only when the coefficients are rational.

For a monic polynomial, the sum of the zeros equals the negative of the coefficient of the next-highest power. If two polynomials of the same degree both vanish at certain zeros, their difference vanishes there too, and its leading coefficient is the difference of the corresponding Vieta sums.

Sign information can be read from Vieta's relations. For ax^2+bx+c=0 with all coefficients positive, the sum -\dfrac{b}{a} is negative and the product \dfrac{c}{a} is positive, so real roots are negative. A claim that a root always has a certain sign, or always lies in a fixed interval, must hold for every admissible choice of coefficients, so one counterexample disproves it.

If a number is a zero of a polynomial, then (x-\text{number}) is a factor. Division leaves a lower-degree polynomial whose zeros are the remaining zeros, and the sum of squares of those zeros follows from their sum and product.

A logarithmic equation \log_{10}E=k means E=10^{k}, which turns it into an algebraic equation. Squaring to remove a square root may introduce extraneous values, so every candidate root must be checked in the original equation.

Simultaneous linear equations (word problems)

A system of simultaneous linear equations determines unknowns uniquely when the number of independent equations equals the number of unknowns. Word problems are solved by naming each unknown, translating each verbal relation into an equation, and then eliminating variables by substitution or addition. Most errors come from setting up the relation incorrectly, so write the sentence in symbols before doing any arithmetic.

When the target quantity is a combination of the given totals, you can avoid solving for each unknown. If the target expression equals \lambda times one equation plus \mu times another, then its value equals \lambda times the first total plus \mu times the second. The multipliers \lambda and \mu are found by matching coefficients.

In age problems, the age of a person at a past or future time is obtained by adding or subtracting the elapsed years from the present age. Every member of a group ages equally, so after t years the total of the ages of m people increases by mt. This rule converts statements about past or future totals into present-time equations.

The same reasoning applies to a family with n children, where the children's total age after t years is nt more than now. The number of children must come out as a whole number, and any solution that violates a stated lower bound on each child's age must be rejected.

Age relations expressed as differences often lead to an absolute value equation, which produces two linear cases. Each case must be solved, and any root giving a negative or non-integer age, where ages must be whole numbers, is discarded.

For a fraction with numerator x and denominator y, each condition becomes a linear relation: the sum of the two parts is given directly, and changing the parts gives a second equation after cross-multiplication. The denominator of the new fraction must be nonzero.

Reward-and-penalty problems model the net amount as a linear expression. If a person earns a fixed amount on each working day and loses a fixed amount on each absent day within a fixed total of days, the net amount is linear in the number of working days, and the equation is solved directly.

A division with a quotient and a remainder obeys the relation dividend = divisor \times quotient + remainder. This converts each statement about a division into a linear equation, and the remainder must be smaller than the divisor, which serves as a check on the answer.

When a quantity is a given percentage of a total, convert the percentage to a fraction. The statement then becomes a linear equation in the two unknowns, and the ratio requested is found after solving the system.

In questions asking whether statements are sufficient, test whether the desired quantity is a combination of the given equations. If either statement alone determines the target, either one suffices. If only the combination of both determines it, both are needed, and if the target is not determined even by the combination, neither suffices.

Nature of roots (discriminant)

For a quadratic ax^2+bx+c=0 with a\neq 0, the discriminant is D=b^2-4ac. If D>0, the roots are real and distinct; if D=0, the roots are real and equal; and if D<0, the roots are non-real complex conjugates. Since D depends only on the coefficients, the nature of the roots can be decided without solving the equation. When the coefficients are rational and D is a perfect square, the roots are rational.

Equal roots require D=0, and the common root is -\dfrac{b}{2a}. For example, x^2-4x+k=0 has equal roots exactly when 16-4k=0.

To identify which equation has no real roots, compute D for each option and look for a negative value. A negative discriminant is the decisive sign that the roots are complex.

If an equation in two variables is treated as a quadratic in one variable, the other variable acts as a parameter, and real solutions demand D\geq 0. When the discriminant is a product containing a factor such as \sin^4\theta-1, and this factor is bounded above by 1, the inequality can hold only at equality. In that case the quadratic becomes a perfect square, (x-y)^2=0, and the variables are forced to be equal.

For real and distinct roots, D>0, and for a parameter k this gives a quadratic inequality in k. Solve by finding the values where D=0 and checking the sign of D on each interval. A result of the form |k|>c corresponds to two separate ranges, one on each side of -c and c.

When the coefficients contain logarithms, the discriminant becomes an inequality in the logarithm. Since \log_{10} is increasing, a bound \log_{10}N\geq c translates into N\geq 10^{c}, so the smallest admissible value of N is found from the bound on the logarithm.

The discriminant can factor. For a parameter-dependent quadratic such as k^2+4k-12, the factorization (k+6)(k-2) gives the sign intervals directly. A quadratic has real linear factors exactly when it has real roots, so the condition for no real linear factors is D<0. The leading coefficient must be checked separately, since if it vanishes the expression is linear and always has a real factor.

If an equation must have a real solution for every value of a parameter \theta in an interval, the discriminant condition must hold throughout that interval. The admissible bound on the other parameter is therefore the minimum of the expression over the interval. Since \cos^2\theta decreases on \left[0,\dfrac{\pi}{4}\right], its minimum occurs at the right endpoint.

Real roots of a quadratic with a parameter occur when D\geq 0, and the greatest value of the parameter is found at the boundary where D=0. When solving the resulting inequality, reverse its direction whenever you divide by a negative number.

Solving quadratic equations

A quadratic equation has the form ax^{2}+bx+c=0 with a\neq 0. Its roots can be found by factorisation, by the quadratic formula x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}, or by a suitable substitution that turns the equation into a simpler one. Whichever method is used, every candidate root must be checked against the original condition, because squaring or clearing denominators can introduce extraneous roots. For example, if the problem states that x is positive, negative roots must be discarded at the end.