Evaluate the discriminant for each equation. Then use it to determine the number and type of solutions. -8x² + 10x = 7

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Rewrite the quadratic equation in standard form \(ax^2 + bx + c = 0\). Start by moving all terms to one side: \(-8x^2 + 10x - 7 = 0\).

Identify the coefficients \(a\), \(b\), and \(c\) from the standard form. Here, \(a = -8\), \(b = 10\), and \(c = -7\).

Recall the formula for the discriminant: \(\Delta = b^2 - 4ac\).

Substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula: \(\Delta = (10)^2 - 4(-8)(-7)\).

Analyze the value of the discriminant to determine the number and type of solutions: if \(\Delta > 0\), there are two distinct real solutions; if \(\Delta = 0\), there is one real repeated solution; if \(\Delta < 0\), there are two complex solutions.

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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 is typically written as ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. To analyze the equation, it must first be rearranged into this standard form by moving all terms to one side.

Discriminant of a Quadratic Equation

The discriminant is given by the formula Δ = b² - 4ac. It helps determine the nature of the roots of a quadratic equation without solving it. The values of the discriminant indicate whether the roots are real and distinct, real and equal, or complex.

Interpreting the Discriminant to Determine Solutions

If the discriminant is positive, the quadratic has two distinct real solutions; if zero, one real repeated solution; and if negative, two complex conjugate solutions. This interpretation guides understanding the number and type of solutions quickly.