Solve each equation using the quadratic formula. $2 x^{2} = 3 - 4 x$

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Rewrite the given equation in standard quadratic form $ax^2 + bx + c = 0$. Start by moving all terms to one side: $2x^2 + 4x - 3 = 0$.

Identify the coefficients $a$, $b$, and $c$ from the quadratic equation. Here, $a = 2$, $b = 4$, and $c = -3$.

Recall the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula will help find the roots of the quadratic equation.

Calculate the discriminant $\Delta = b^2 - 4ac$ by substituting the values of $a$, $b$, and $c$: $\Delta = 4^2 - 4 \times 2 \times (-3)$.

Substitute the values of $a$, $b$, and the discriminant $\Delta$ into the quadratic formula to express the solutions for $x$: $x = \frac{-4 \pm \sqrt{\Delta}}{2 \times 2}$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 depending on the discriminant.

Discriminant and Nature of Roots

The discriminant, given by b² - 4ac, determines the type of 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. This helps predict the solution's nature before calculation.

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