Questions: Find the zeros of the function. Give exact solutions. x^2 + 7x - 5 = 0 The exact solutions are x = (Simplify your answer. Use integers or f as needed.)

Transcript text: Find the zeros of the function. Give exact solutions. \[ x^{2}+7 x-5=0 \] The exact solutions are $x=$ $\square$ (Simplify your answer. Use integers or $f$ as needed.)

Solution

Solution Steps

To find the zeros of the quadratic function $x^2 + 7x - 5 = 0$, we can use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation $ax^2 + bx + c = 0$.

Step 1: Identify the Coefficients

Given the quadratic equation $x^2 + 7x - 5 = 0$, we identify the coefficients:

$a = 1$
$b = 7$
$c = -5$

Step 2: Calculate the Discriminant

The discriminant $\Delta$ of the quadratic equation is given by: \[ \Delta = b^2 - 4ac \] Substituting the values of $a$, $b$, and $c$: \[ \Delta = 7^2 - 4 \cdot 1 \cdot (-5) = 49 + 20 = 69 \]

Step 3: Apply the Quadratic Formula

The solutions to the quadratic equation are given by: \[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \] Substituting the values of $a$, $b$, and $\Delta$: \[ x = \frac{-7 \pm \sqrt{69}}{2 \cdot 1} \]

Step 4: Calculate the Exact Solutions

We calculate the two solutions: \[ x_1 = \frac{-7 + \sqrt{69}}{2} \approx 0.6533 \] \[ x_2 = \frac{-7 - \sqrt{69}}{2} \approx -7.6533 \]