In Exercises 75–82, compute the discriminant. Then determine the number and type of solutions for the given equation. $2 x^{2} - 11 x + 3 = 0$

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Identify the coefficients in the quadratic equation $2x^2 - 11x + 3 = 0$. Here, $a = 2$, $b = -11$, and $c = 3$.

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

Substitute the values of $a$, $b$, and $c$ into the discriminant formula: $\Delta = (-11)^2 - 4 \times 2 \times 3$.

Simplify the expression to find the value of the discriminant (do not calculate the final number yet).

Use 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 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

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 its coefficients.

Discriminant

The discriminant is the part of the quadratic formula under the square root, given by b² - 4ac. It determines the nature and number of solutions of a quadratic equation: if positive, two distinct real solutions; if zero, one real repeated solution; if negative, two complex solutions.

Types of Solutions of Quadratic Equations

The solutions of a quadratic equation can be real or complex. Based on the discriminant, the equation may have two distinct real roots, one repeated real root, or two complex conjugate roots. Understanding these types helps in interpreting the behavior of the quadratic function.

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Textbook Question

The rule for rewriting an absolute value equation without absolute value bars can be extended to equations with two sets of absolute value bars: If u and v represent algebraic expressions, then |u| = |v| is equivalent to u = v or u = - v. Use this to solve the equations in Exercises 77–84. |4x - 3| = |4x - 5|

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Textbook Question

In Exercises 59–94, solve each absolute value inequality. |3 - (2/3)x| > 5

In Exercises 59–94, solve each absolute value inequality. 3|x - 1| + 2 ≥ 8

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