Solve by Graphing

Quadratic Roots and Coefficients on the SAT: Desmos Walkthrough

Pick easy numbers for the unknowns, graph the quadratic, click its

x

-intercepts, and let Desmos confirm the relationship between the roots and the coefficients.

This lesson solves by graphing: turn the equation into curves, then read intersections, vertices, or tangency straight off the Desmos graph.

Take this lesson interactively

A real SAT-style question

$57 x^{2} + (57 b + a) x + a b = 0$

In the given equation, $a$ and $b$ are positive constants. The product of the solutions to the given equation is $k a b$ , where $k$ is a constant. What is the value of $k$ ?

$\frac{1}{57}$

$\frac{1}{19}$

$1$

$57$

How to solve it in Desmos, step by step

The idea

Pick easy numbers for a and b, graph the quadratic as a function, click its two x-intercepts, then have Desmos compute the product of the solutions and compare it to a times b. You will see the product equals a times b divided by 57.

1
Step 1
Set a and b to easy numbers. Type these two lines so the equation has real numbers in it.
Type in Desmos
$a=2$ $b=3$
2
Step 2
Graph the quadratic as a function of x. Type this line and Desmos draws a parabola that crosses the x-axis in two places.
Type in Desmos
$y=57x^{2}+(57b+a)x+ab$
3
Step 3
Hover over each spot where the curve crosses the x-axis and click the gray dot. Desmos labels the two solutions: one near x equals negative 3 and one near x equals negative 0.0351.
4
Step 4
Multiply the two solutions. Type this line and Desmos displays 0.105263 as the product.
Type in Desmos
$\left(-3\right)\cdot\left(-\frac{2}{57}\right)$
5
Step 5
Compare against a times b divided by 57. Type this line and Desmos also displays 0.105263, the same value. So the product equals a times b divided by 57, which means k is 1 over 57. The answer is A.
Type in Desmos
$\frac{ab}{57}$

The answer is A.

$\frac{1}{57}$

Now you try one

Same question type, different numbers. Use the exact Desmos moves from the walkthrough above.

23x^2 + (23b + a)x + ab = 0

In the given equation,

a

and

b

are positive constants. The product of the solutions to the given equation is

kab

, where

k

is a constant. What is the value of

k

Practice this with a live Desmos calculator

The interactive version of this lesson builds the Desmos graph up step by step as you click through, with a live calculator you can type into and your progress saved.

Open the interactive lesson