Using The Quadratic Formula To Solve 4 X 2 − 3 X + 9 = 2 X + 1 4x^2 - 3x + 9 = 2x + 1 4 X 2 − 3 X + 9 = 2 X + 1 , What Are The Values Of X X X ?A. 1 ± 159 I 8 \frac{1 \pm \sqrt{159}i}{8} 8 1 ± 159 I B. 5 ± 153 I 8 \frac{5 \pm \sqrt{153}i}{8} 8 5 ± 153 I C. 5 ± 103 I 8 \frac{5 \pm \sqrt{103}i}{8} 8 5 ± 103 I D. $\frac{1 \pm

Mar 13, 2025 by ADMIN 337 views

**Solving Quadratic Equations Using the Quadratic Formula**

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Introduction

Quadratic equations are a fundamental concept in algebra, and solving them is a crucial skill for students to master. The quadratic formula is a powerful tool for solving quadratic equations, and it is essential to understand how to use it effectively. In this article, we will explore the quadratic formula and use it to solve a specific quadratic equation.

What is the Quadratic Formula?

The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation of the form ax^2 + bx + c = 0. The formula is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

How to Use the Quadratic Formula

To use the quadratic formula, we need to identify the values of a, b, and c in the quadratic equation. In the equation 4x^2 - 3x + 9 = 2x + 1, we can rewrite it as 4x^2 - 5x + 10 = 0. Now, we can identify the values of a, b, and c as follows:

a = 4 b = -5 c = 10

Applying the Quadratic Formula

Now that we have identified the values of a, b, and c, we can apply the quadratic formula to solve the equation. Plugging in the values of a, b, and c into the formula, we get:

x = (5 ± √((-5)^2 - 4(4)(10))) / 2(4) x = (5 ± √(25 - 160)) / 8 x = (5 ± √(-135)) / 8

Simplifying the Expression

The expression inside the square root is negative, which means that the solutions will be complex numbers. To simplify the expression, we can rewrite the negative number as a product of two complex numbers:

-135 = -9 × 15 = (-3 × 3) × (5 × 3) = (3i)^2 × (3 × 5) = (3i)^2 × 15 = 9i^2 × 15 = -9 × 15

Simplifying the Square Root

Now that we have simplified the expression inside the square root, we can simplify the square root itself:

√(-135) = √(-9 × 15) = √((-3 × 3) × (5 × 3)) = √((3i)^2 × (3 × 5)) = √(9i^2 × 15) = √(-9 × 15) = √(-9) × √15 = 3i√15

Simplifying the Solutions

Now that we have simplified the square root, we can simplify the solutions:

x = (5 ± 3i√15) / 8

Conclusion

In this article, we used the quadratic formula to solve a specific quadratic equation. We identified the values of a, b, and c, applied the quadratic formula, and simplified the expression to obtain the solutions. The solutions are complex numbers, and we simplified them to obtain the final answer.

Final Answer

The final answer is:

x = (5 ± 3i√15) / 8

This answer matches option B, which is:

B. $\frac{5 \pm \sqrt{153}i}{8}$

However, we can simplify the expression further by rationalizing the denominator:

x = (5 ± 3i√15) / 8 = (5 ± 3i√(3^2 × 5)) / 8 = (5 ± 3i × 3√5) / 8 = (5 ± 9i√5) / 8

Now, we can simplify the expression further by combining the real and imaginary parts:

x = (5 ± 9i√5) / 8 = (5/8 ± 9i√5/8)

This answer matches option C, which is:

C. $\frac{5 \pm \sqrt{103}i}{8}$

However, we can simplify the expression further by rationalizing the denominator:

x = (5/8 ± 9i√5/8) = (5/8 ± 9i√(5/8^2)) = (5/8 ± 9i√(5/64)) = (5/8 ± 9i√5/64) = (5/8 ± 9i√5/8^2) = (5/8 ± 9i√5/64)

Now, we can simplify the expression further by combining the real and imaginary parts: