Let's solve the quadratic equation [tex]\(7x^2 = 9 + x\)[/tex] step-by-step to identify which equation shows the quadratic formula used correctly to solve for [tex]\(x\)[/tex].
1. Rewrite the equation in standard form:
Given [tex]\(7x^2 = 9 + x\)[/tex], move all terms to one side to set the equation to 0.
[tex]\[
7x^2 - x - 9 = 0
\][/tex]
2. Identify the coefficients:
The quadratic equation is now in the form [tex]\(ax^2 + bx + c = 0\)[/tex], where:
[tex]\[
a = 7, \quad b = -1, \quad c = -9
\][/tex]
3. Quadratic formula:
The quadratic formula to solve [tex]\(ax^2 + bx + c = 0\)[/tex] is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
4. Substitute the coefficients:
Substitute [tex]\(a = 7\)[/tex], [tex]\(b = -1\)[/tex], and [tex]\(c = -9\)[/tex] into the quadratic formula.
[tex]\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(7)(-9)}}{2(7)}
\][/tex]
5. Simplify the expression:
[tex]\[
x = \frac{1 \pm \sqrt{1 - 4 \cdot 7 \cdot (-9)}}{2 \cdot 7}
\][/tex]
[tex]\[
x = \frac{1 \pm \sqrt{1 + 252}}{14}
\][/tex]
[tex]\[
x = \frac{1 \pm \sqrt{253}}{14}
\][/tex]
Given the steps above, let's evaluate the provided options to see which one matches:
1. [tex]\(x = \frac{-1 \pm \sqrt{(1)^2-4(7)(9)}}{2(7)}\)[/tex]
2. [tex]\(x = \frac{1 \pm \sqrt{(-1)^2-4(7)(9)}}{2(7)}\)[/tex]
- This simplifies correctly based on our derivation, as it correctly uses the values [tex]\(a = 7\)[/tex], [tex]\(b = -1\)[/tex], and [tex]\(c = -9\)[/tex].
3. [tex]\(x = \frac{-1 \pm \sqrt{(-1)^2+4(7)(9)}}{2(7)}\)[/tex]
4. [tex]\(x = \frac{1 \pm \sqrt{(-1)^2+4(7)(9)}}{2(7)}\)[/tex]
The correct equation showing the quadratic formula used correctly to solve [tex]\(7x^2 = 9 + x\)[/tex] for [tex]\(x\)[/tex] is:
[tex]\[
x = \frac{1 \pm \sqrt{(-1)^2-4(7)(9)}}{2(7)}
\][/tex]
