Solving the Equation 4x^2 + 5x – 12 = 0

Solution. 4 x ²−5 x −12=0 : x =5+√2178 , x =5−√2178 ( Decimal : x =2. 46636…, x =−1. 21636…)

Introduction

Quadratic equations, like the one given in the keyword 4x^2 + 5x – 12 = 0 are polynomial equations of degree two. These equations can be solved using various methods, including factoring, completing the square, and the quadratic formula. This article will guide you through solving the quadratic equation 4×2+5x−12=04x^2 + 5x – 12 = 04×2+5x−12=0 step by step.

Quadratic Equations

General Form

A quadratic equation is generally written as: ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 where aaa, bbb, and ccc are constants, and xxx represents the variable.

Identifying Coefficients

For the equation 4×2+5x−12=04x^2 + 5x – 12 = 04×2+5x−12=0:

a=4a = 4a=4
b=5b = 5b=5
c=−12c = -12c=−12

Methods of Solving Quadratic Equations

Method 1: Factoring

Factoring involves expressing the quadratic equation as a product of two binomials. This method is straightforward if the equation can be easily factored.

Equation: 4×2+5x−12=04x^2 + 5x – 12 = 04×2+5x−12=0
Finding Factors: Look for two numbers that multiply to a⋅c=4⋅(−12)=−48a \cdot c = 4 \cdot (-12) = -48a⋅c=4⋅(−12)=−48 and add to b=5b = 5b=5.

The factors of -48 that add up to 5 are 8 and -6.

Rewrite the Equation: Express 5x5x5x as the sum of these factors: 4×2+8x−6x−12=04x^2 + 8x – 6x – 12 = 04×2+8x−6x−12=0
Factor by Grouping: Group terms and factor each group:

4x(x+2)−6(x+2)=04x(x + 2) – 6(x + 2) = 04x(x+2)−6(x+2)=0 (4x−6)(x+2)=0(4x – 6)(x + 2) = 0(4x−6)(x+2)=0

Solve for x: Set each factor equal to zero:

4x−6=0orx+2=04x – 6 = 0 \quad \text{or} \quad x + 2 = 04x−6=0orx+2=0 x=64=32orx=−2x = \frac{6}{4} = \frac{3}{2} \quad \text{or} \quad x = -2x=46=23orx=−2

Method 2: Quadratic Formula

The quadratic formula can solve any quadratic equation: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}x=2a−b±b2−4ac

Equation: 4×2+5x−12=04x^2 + 5x – 12 = 04×2+5x−12=0
Coefficients: a=4a = 4a=4, b=5b = 5b=5, c=−12c = -12c=−12
Discriminant: Calculate b2−4acb^2 – 4acb2−4ac:

b2−4ac=52−4(4)(−12)=25+192=217b^2 – 4ac = 5^2 – 4(4)(-12) = 25 + 192 = 217b2−4ac=52−4(4)(−12)=25+192=217

Apply the Quadratic Formula:

x=−5±2178x = \frac{-5 \pm \sqrt{217}}{8}x=8−5±217

Method 3: Completing the Square

Completing the square involves transforming the quadratic equation into a perfect square trinomial.

Equation: 4×2+5x−12=04x^2 + 5x – 12 = 04×2+5x−12=0
Divide by a: Ensure the coefficient of x2x^2×2 is 1:

x2+54x−3=0x^2 + \frac{5}{4}x – 3 = 0x2+45x−3=0

Move the constant term:

x2+54x=3x^2 + \frac{5}{4}x = 3×2+45x=3

Complete the Square: Add and subtract (b2)2\left(\frac{b}{2}\right)^2(2b)2:

x2+54x+(58)2=3+(58)2x^2 + \frac{5}{4}x + \left(\frac{5}{8}\right)^2 = 3 + \left(\frac{5}{8}\right)^2×2+45x+(85)2=3+(85)2

Simplify and solve:

(x+58)2=21764\left(x + \frac{5}{8}\right)^2 = \frac{217}{64}(x+85)2=64217 x+58=±21764x + \frac{5}{8} = \pm \sqrt{\frac{217}{64}}x+85=±64217 x=−58±2178x = -\frac{5}{8} \pm \frac{\sqrt{217}}{8}x=−85±8217

Conclusion

Solving the quadratic equation 4×2+5x−12=04x^2 + 5x – 12 = 04×2+5x−12=0 can be done using different methods. Factoring gives us the solutions x=32x = \frac{3}{2}x=23 and x=−2x = -2x=−2. The quadratic formula confirms these results, offering a reliable way to find the roots. Completing the square provides a more visual and step-by-step method, also leading to the same solutions.

FAQs

1. What is a quadratic equation?

A quadratic equation is a second-degree polynomial equation in the form ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0.

2. How do you solve a quadratic equation by factoring?

To solve by factoring, express the quadratic equation as a product of two binomials and set each factor to zero to solve for xxx.

3. What is the quadratic formula?

The quadratic formula is x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}x=2a−b±b2−4ac, used to find the roots of any quadratic equation.

4. What does completing the square mean?

Completing the square is a method of solving quadratic equations by converting them into a perfect square trinomial.

5. Can all quadratic equations be factored easily?

Not all quadratic equations can be factored easily. In such cases, the quadratic formula or completing the square are more reliable methods.

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