Ch. R - Algebra Review

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. R - Algebra Review

Problem R.6.69

Chapter 1, Problem R.6.69

Solve each quadratic equation using the quadratic formula. See Example 7. -2x² + 4x + 3 = 0

Verified step by step guidance

Identify the coefficients from the quadratic equation \(-2x^2 + 4x + 3 = 0\). Here, \(a = -2\), \(b = 4\), and \(c = 3\).

Recall the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula gives the solutions for any quadratic equation \(ax^2 + bx + c = 0\).

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

Evaluate the square root of the discriminant \(\sqrt{\Delta}\) to determine the nature of the roots (real and distinct, real and equal, or complex).

Substitute \(b\), \(\sqrt{\Delta}\), and \(a\) into the quadratic formula to write the two possible 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

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 real and complex solutions.

Discriminant

The discriminant, given by Δ = b² - 4ac, determines the nature of the roots of a quadratic equation. If Δ > 0, there are two distinct real roots; if Δ = 0, one real root; and if Δ < 0, two complex conjugate roots.

Related Practice

Textbook Question

Solve each quadratic equation using the zero-factor property. See Example 5. -4x² + x = -3

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Textbook Question

Solve each inequality. Give the solution set using interval notation. See Example 10.

- 4 ≤ (x + 1)/2 ≤ 5

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Textbook Question

Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. -3(x - 6) > 2x - 2

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Textbook Question

Determine whether each equation is an identity, a conditional equation, or a contradiction. Give the solution set. See Example 4. -2(x + 3) = -6(x + 7)

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Textbook Question

Solve each quadratic equation using the square root property. See Example 6. (3x - 1)² = 12

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Textbook Question

CONCEPT PREVIEW Perform the indicated operation, and write each answer in lowest terms 3 7 —— + —— x x

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