Solve each equation in Exercises 65–74 using the quadratic formula. $4 x^{2} = 2 x + 7$

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Rewrite the equation in standard quadratic form $ax^2 + bx + c = 0$. Start by moving all terms to one side: $4x^2 - 2x - 7 = 0$.

Identify the coefficients: $a = 4$, $b = -2$, and $c = -7$.

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: $x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(4)(-7)}}{2(4)}$.

Simplify inside the square root and the numerator to prepare for solving: calculate the discriminant $b^2 - 4ac$ and then write the expression for $x$ before finding the final values.

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주요 개념

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Quadratic Equation Standard Form

A quadratic equation must be written in the standard form ax² + bx + c = 0 before applying the quadratic formula. This involves rearranging all terms to one side of the equation so that the other side equals zero, allowing identification of coefficients a, b, and c.

Quadratic Formula

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

Discriminant and Nature of Roots

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. 4x2=2x+74x^2 = 2x + 7