Point OnePrep

SAT Math: Quadratic From a Table — Find the Coefficient b

Hard Digital SAT Advanced Math grid-in. A table gives the zeros of y = 24x² − bx − 224; find b. Includes a Vieta shortcut and a Desmos regression solution.

Question

xxyy
−43-\dfrac{4}{3}0
0-224
70

The table shows three values of xx and their corresponding values of yy. There is a quadratic relationship between xx and yy. An equation that represents this relationship can be written as y=24x2−bx−224y = 24x^{2} - bx - 224, where bb is a constant. What is the value of bb?

Step-by-Step Solution

Read the zeros from the table, then plug back into the equation.

1Read the zeros off the table.

The two rows where y = 0 give the x-values that make the quadratic equal to zero — i.e., its zeros (roots):

x=−43andx=7x = -\dfrac{4}{3} \qquad \text{and} \qquad x = 7

The third row (x = 0, y = −224) just confirms the constant term and isn’t needed to find b.

2Plug a zero back into the equation.

Use x = 7 because it avoids fractions. Setting y = 0 in y = 24x² − bx − 224:

0=24(7)2−b(7)−2240 = 24(7)^{2} - b(7) - 224
0=24(49)−7b−2240 = 24(49) - 7b - 224
0=1176−7b−224=952−7b0 = 1176 - 7b - 224 = 952 - 7b
7b=952  ⟹  b=1367b = 952 \;\Longrightarrow\; b = 136

3Shortcut: Vieta’s formulas.

For a quadratic ax² + Bx + c with zeros r₁ and r₂, the sum of the zeros is −B/a. Here the linear coefficient is −b, so:

r1+r2=−−b24=b24r_{1} + r_{2} = -\dfrac{-b}{24} = \dfrac{b}{24}

The sum of the zeros from the table is:

−43+7=−43+213=173-\dfrac{4}{3} + 7 = -\dfrac{4}{3} + \dfrac{21}{3} = \dfrac{17}{3}

Set them equal and solve:

b24=173  ⟹  b=24⋅173=8⋅17=136\dfrac{b}{24} = \dfrac{17}{3} \;\Longrightarrow\; b = 24 \cdot \dfrac{17}{3} = 8 \cdot 17 = 136

4Calculator path: Desmos quadratic regression.

In Desmos, type "table" and enter the three (x, y) pairs from the question. Then add a regression line:

y1∼a x12+B x1+cy_{1} \sim a\,x_{1}^{2} + B\,x_{1} + c

Desmos returns a ≈ 24, B ≈ −136, c ≈ −224. The question writes the quadratic as y = 24x² − bx − 224, so its linear coefficient is −b. Matching:

−b=−136  ⟹  b=136-b = -136 \;\Longrightarrow\; b = 136

Watch the sign — the b you submit is the negative of the B Desmos prints. This is exactly the trap.

5Sanity check.

Verify b = 136 by plugging x = −4/3 back into the equation:

24(169)−136(−43)−224=3849+5443−224=1283+5443−22424\left(\tfrac{16}{9}\right) - 136\left(-\tfrac{4}{3}\right) - 224 = \tfrac{384}{9} + \tfrac{544}{3} - 224 = \tfrac{128}{3} + \tfrac{544}{3} - 224
=6723−224=224−224=0  ✓= \tfrac{672}{3} - 224 = 224 - 224 = 0 \;\checkmark

Both zeros satisfy the equation, so b = 136.