Question type

Quadratic formula questions on the SAT

Use the quadratic formula, $x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$ , when the SAT asks for a root in exact form with a square root, or works backwards from a given root to a constant. Desmos gives the root as a decimal, so match it against the choices or use the formula.

These questions give a quadratic whose roots are not whole numbers and ask for a root in exact radical form, such as $\frac{6 + 46}{5}$ , or give a root and ask for a constant in the equation.

It appeared on two of the five official practice tests we analysed.

On 2 of 5 official practice tests
22 practice questions of this type
Desmos partly helps

Why students miss it

Sign errors under the root: $b^{2} - 4 a c$ with a negative $a$ or $c$ becomes $b^{2}$ plus something. For $- 5 t^{2} + 12 t + 2 = 0$ , it is $144 + 40$ , not $144 - 40$ .

Dividing only part of the expression by $2 a$ . The whole numerator, $- b \pm \dots$ , sits over $2 a$ , and simplifying $\frac{12 \pm 2 46}{10}$ means dividing every term by 2.

Picking the wrong root. A height that reaches zero, or a positive length, needs the positive root.

How to spot it

Answer choices written with a square root, such as $\frac{a \pm b}{c}$ .
A quadratic that does not factor with whole numbers.
"One solution to the equation is ...", followed by a radical and a missing constant.

The method

Exact roots

Write the equation as $a x^{2} + b x + c = 0$ before reading off $a$ , $b$ and $c$ .

Work out $b^{2} - 4 a c$ first: and simplify the root.
Put the whole of $- b \pm \dots$ over $2 a$: then cancel common factors.
Context needs one root: keep the positive or larger one.

Given a root, find a constant

Write the formula with the unknown constant left in it.

Match the part outside the root: this often fixes $b$ or $a$ .
Match the part under the root: this gives $c$ .

With Desmos

Desmos finds the root as a decimal. That is enough only if you then match it to a choice.

Type the quadratic, for example y=-5x^2+12x+2, and click the x-intercept to read its decimal value.
Type each answer choice as a number, such as (6+sqrt(46))/5, and see which one equals the intercept.

When the choices are formulas in a constant, as in "which gives the solutions to $a x^{2} - 4 x - 3 = 0$ ", there is nothing to graph. Use the formula.

Two real questions

From the 1500prep bank, the way the app shows them. Answer first, then open the explanation.

1Question 1 of 2#705Adv

The height, in meters, of a ball $t$ seconds after it is thrown is modeled by $h = - 5 t^{2} + 12 t + 2$ . For what value of $t$ does the ball reach the ground, where $h = 0$ ?

Pick an answer, then check it.

Show the answer and explanation

Solve

- 5 t^{2} + 12 t + 2 = 0

, or equivalently

5 t^{2} - 12 t - 2 = 0

t = \frac{12 \pm 144 + 40}{10} = \frac{12 \pm 184}{10}

. Since

184 = 246

, this is

\frac{6 \pm 46}{5}

. Time must be positive, so take the plus sign.

2Question 2 of 2#735Adv

One solution to the equation $4 x^{2} + 4 x + c = 0$ is $\frac{- 1 + 6}{2}$ , where $c$ is a constant. What is the value of $c$ ?

Pick an answer, then check it.

Show the answer and explanation

Match the given root to the quadratic formula. For

4 x^{2} + 4 x + c = 0

x = \frac{- 4 \pm 16 - 16 c}{8} = \frac{- 1 \pm 1 - c}{2}

. Since the root is

\frac{- 1 + 6}{2}

, you need

1 - c = 6

, so

c = - 5

. Check with the product of roots:

\frac{- 1 + 6}{2} \cdot \frac{- 1 - 6}{2} = \frac{1 - 6}{4} = - \frac{5}{4} = \frac{c}{4}

, giving

c = - 5

Questions students ask

When should I use the quadratic formula on the SAT?

When the quadratic does not factor neatly and the answer must be exact, with a square root in it, or when you are given a root and asked for a constant in the equation.

Is the quadratic formula on the SAT reference sheet?

No. The reference sheet gives area and volume formulas and special right triangles, not the quadratic formula. You need to know it.

Can Desmos solve quadratic formula questions?

Desmos gives the roots as decimals. Type each answer choice into Desmos to see which matches. When the choices contain a letter, use the formula itself.

Practise this type until it is automatic

Five more quadratic-formula questions, checked instantly, with every wrong choice explained. Free, no card.

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