Evaluate the discriminant for each equation. Then use it to determine the number and type of solutions. 16x² +3 = -26x

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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: \(16x^2 + 26x + 3 = 0\).

Identify the coefficients: \(a = 16\), \(b = 26\), and \(c = 3\) from the standard form.

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

Substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula: \(\Delta = (26)^2 - 4 \times 16 \times 3\).

Analyze the value of the discriminant: if \(\Delta > 0\), there are two distinct real solutions; if \(\Delta = 0\), there is exactly one real 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. To analyze or solve 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 value indicates 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; if negative, two complex conjugate solutions. This interpretation guides understanding the number and type of solutions.