Sum and Product of Roots | SAT Math Practice

The key idea

For

a x^{2} + b x + c = 0

, the solutions add to

- \frac{b}{a}

and multiply to

\frac{c}{a}

. These give quick routes to sums, products, a missing root, and expressions like

p^{2} + q^{2} = (p + q)^{2} - 2 pq

Using $\frac{b}{a}$ for the sum; the sign is negative.

Desmos finds the roots of a known equation, but the shortcut is faster, and it is the only route when a coefficient is unknown.

College Board skill: Nonlinear equations in one variable and systems of equations in two variables · Advanced Math

One question from each level. The drill has 12 at every level, and moves you up after four right in a row.

Level 1 · Warm-up

What is the sum of the solutions to

3 x^{2} - 16 x - 13 = 0

Show the answer

Answer: 16/3Since

c < 0

and

a > 0

, the equation has two real solutions. Their sum is

- \frac{b}{a} = - \frac{- 16}{3} = \frac{16}{3}

Level 2 · Test level

In the equation

4 x^{2} + k x - 72 = 0

k

is a constant. If one solution is

- 9

, what is the other solution?

Show the answer

Answer: 2The product of the solutions is

\frac{c}{a} = - 18

. With one solution

- 9

, the other is

- 18 \div (- 9) = 2

. You don't need

k

Level 3 · Beyond the test

The solutions to

x^{2} - 10 x - 9 = 0

are

p

and

q

. What is the value of

p^{2} + q^{2}

Show the answer

Answer: 118

p + q = 10

and

pq = - 9

. Then

p^{2} + q^{2} = (p + q)^{2} - 2 pq = 100 - 2 (- 9) = 118

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