Question

modeling and analyzing quadratic functions: quadratic equations zacharias is using the quadratic formula to solve the equation 0 = -2x² + 5x - 3. he begins by substituting as shown. quadratic formula: ( x = \frac{-b pm sqrt{b^2 - 4ac}}{2a} ) substitution: ( x = \frac{-5 pm sqrt{5^2 - 4(2)(-3)}}{2(-2)} ) what error did zacharias make? the (-5) should be (5). the (5^2) should be (-5^2). the (2) in the numerator should be (-2). the (2) in the denominator should be (-2).

Explanation:

Step1: Identify standard quadratic form

A quadratic equation is $ax^2+bx+c=0$. For $0=-2x^2+5x-3$, we have $a=-2$, $b=5$, $c=-3$.

Step2: Recall quadratic formula

The formula is $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Step3: Substitute values correctly

Substitute $a,b,c$: $x=\frac{-(5)\pm\sqrt{5^2-4(-2)(-3)}}{2(-2)}=\frac{-5\pm\sqrt{25-24}}{-4}$

Step4: Compare with Zacharias' work

Zacharias wrote $x=\frac{-5\pm\sqrt{5^2-4(2)(-3)}}{2(-2)}$. He used $2$ instead of $-2$ for $a$ inside the discriminant, but the error in the options matches: the $-5$ is correct, $5^2$ is correct, denominator $2(-2)$ is correct. Wait, no—wait, the numerator's $-b$ is $-5$, which is correct. Wait, no: wait, the error is that in the discriminant, he used $4(2)(-3)$ instead of $4(-2)(-3)$, but the options given: the correct error is that the $2$ in the numerator? No, wait no: wait, no, let's recheck. Wait, the quadratic formula is $-b\pm\sqrt{b^2-4ac}$. $b=5$, so $-b=-5$, that is correct. $b^2=5^2$, correct. $4ac$ is $4*(-2)*(-3)$, Zacharias wrote $4(2)(-3)$, so he used positive 2 instead of -2 for $a$ in the discriminant. But looking at the options, the option "The 2 in the numerator should be -2" is not that. Wait no, wait the options:
Wait, no, wait the substitution Zacharias did: $x=\frac{-5\pm\sqrt{5^2-4(2)(-3)}}{2(-2)}$. The $4(2)(-3)$ is wrong, it should be $4(-2)(-3)$. But the options given: the only one that aligns is that the 2 inside the square root (the $a$ value) is wrong, but the option says "The 2 in the numerator should be -2" no, wait no. Wait, no, wait I misread. Wait, the options:

1. The -5 should be 5: no, $-b=-5$, correct.
2. The $5^2$ should be $-5^2$: no, $b^2=5^2$, correct.
3. The 2 in the numerator should be -2: no, numerator is $-5\pm...$, no. Wait, no, wait the $4ac$ term: $4*a*c=4*(-2)*(-3)$, Zacharias wrote $4*(2)*(-3)$, so he used $a=2$ instead of $a=-2$ in the discriminant. But the option says "The 2 in the denominator should be -2": denominator is $2a=2*(-2)$, which is correct. Wait, no, wait the problem is: Zacharias' substitution is $x=\frac{-5\pm\sqrt{5^2-4(2)(-3)}}{2(-2)}$. The error is that in the discriminant, he used $2$ instead of $-2$ for $a$. But the options: the only possible correct option is that the 2 in the numerator (inside the square root) should be -2? Wait no, the option says "The 2 in the numerator should be -2". Oh, right, the $4(2)(-3)$ is part of the numerator's square root. So that 2 should be -2.

Wait, no, let's re-express:
Correct substitution: $x=\frac{-5\pm\sqrt{5^2-4(-2)(-3)}}{2(-2)}$
Zacharias' substitution: $x=\frac{-5\pm\sqrt{5^2-4(2)(-3)}}{2(-2)}$
So the error is that the 2 inside the square root (the $a$ value) is positive instead of negative, which matches the option "The 2 in the numerator should be -2".

Answer:

The 2 in the numerator should be -2.