Textbook Question
Solve each equation. 2x2+x-15 = 0
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1. Equations & Inequalities
The Square Root Property
1:40 minutesProblem 51
Textbook Question
Evaluate the discriminant for each equation. Then use it to determine the number and type of solutions. -8x² + 10x = 7
Verified step by step guidance1
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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Video duration:
1mKey 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.
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Converting Standard Form to Vertex Form
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.
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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.
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