Solve each equation in Exercises 65–74 using the quadratic formula. $3 x^{2} - 3 x - 4 = 0$

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Identify the coefficients in the quadratic equation $3x^2 - 3x - 4 = 0$. Here, $a = 3$, $b = -3$, and $c = -4$.

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

Calculate the discriminant $\Delta = b^2 - 4ac$ by substituting the values: $\Delta = (-3)^2 - 4(3)(-4)$.

Substitute $a$, $b$, and the discriminant $\Delta$ into the quadratic formula: $x = \frac{-(-3) \pm \sqrt{\Delta}}{2(3)}$.

Simplify the expression under the square root and the entire fraction to express the two possible solutions for $x$.

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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, b, and c are constants and a ≠ 0. It represents a parabola when graphed and can have zero, one, or two real solutions depending on the discriminant.

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 find the roots, including complex solutions when the discriminant is negative.

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. 3x2−3x−4=03x^2 - 3x - 4 = 0