Solve each equation in Exercises 65–74 using the quadratic formula. $x^{2} + 8 x + 15 = 0$

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Identify the coefficients in the quadratic equation $x^2 + 8x + 15 = 0$. Here, $a = 1$, $b = 8$, and $c = 15$.

Recall the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Substitute the values of $a$, $b$, and $c$ into the quadratic formula: $x = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 1 \cdot 15}}{2 \cdot 1}$.

Simplify inside the square root (the discriminant): calculate $8^2 - 4 \cdot 1 \cdot 15$ to determine the value under the square root.

Evaluate the square root and then compute the two possible values for $x$ by using the plus and minus signs in the formula.

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주요 개념

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Quadratic Equation

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a ≠ 0. It represents a parabola when graphed, and its solutions are the x-values where the parabola intersects the x-axis.

Quadratic Formula

The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) provides the solutions to any quadratic equation ax² + bx + c = 0. It uses coefficients a, b, and c to find roots, including real and complex solutions.

Discriminant

The discriminant, given by b² - 4ac, determines the nature of the roots of a quadratic equation. If positive, there are two distinct real roots; if zero, one real root; and if negative, two complex conjugate roots.

Solve each equation in Exercises 65–74 using the quadratic formula. x2+8x+15=0x^2 + 8x + 15 = 0