Find the zeros of the function. Give exact answers and approximate solutions rounded to three decimal places when possible.
$x^{2} + 7 x - 3 = 0$
The exact solutions are $x =$
(Simplify your answer Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals and $i$ as needed. Use a comma to separate answers as needed )
The approximate solutions to three decimal places are $x \approx$
(Use a comma to separate answers as needed.)

Answer

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Answer

$Approximate solutions: x \approx 0.405, x \approx - 7.405$

Steps

Step 1 ：Given the quadratic equation $x^{2} + 7 x - 3 = 0$ , we can use the quadratic formula to find the exact solutions:

Step 2 ： $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

Step 3 ：Here, $a = 1$ , $b = 7$ , and $c = - 3$ .

Step 4 ：Calculate the discriminant: $Δ = b^{2} - 4 a c = 7^{2} - 4 (1) (- 3) = 61$

Step 5 ：Plug the values into the quadratic formula: $x = \frac{- 7 \pm \sqrt{61}}{2}$

Step 6 ：The exact solutions are $x = \frac{- 7 + \sqrt{61}}{2}$ and $x = \frac{- 7 - \sqrt{61}}{2}$

Step 7 ：Approximate the solutions to three decimal places: $x \approx 0.405$ and $x \approx - 7.405$

Step 8 ： $Exact solutions: x = \frac{- 7 + \sqrt{61}}{2}, x = \frac{- 7 - \sqrt{61}}{2}$

Step 9 ： $Approximate solutions: x \approx 0.405, x \approx - 7.405$