Solve Quadratic Equation (4x 2 5x 12 0)

Understanding and solving quadratic equations is a fundamental part of algebra that plays a significant role in both academic and real-life applications. Among the wide range of quadratic equations, one commonly encountered example is 4x 2 5x 12 0, which, when formatted correctly, represents the equation 4x² + 5x – 12 = 0. This equation may look complex at first glance, especially to students just beginning their journey in algebra, but with the right approach, it can be solved efficiently using various mathematical techniques. In this article, we will explore what a quadratic equation is, break down the components of the given equation, and go through different methods of solving it with detailed explanations.

What Is a Quadratic Equation?

A quadratic equation is a second-degree polynomial equation in a single variable x with the general form:

ax² + bx + c = 0

Here:

• a is the coefficient of x²,

• b is the coefficient of x,

• c is the constant term.

The key feature of a quadratic equation is that the highest power of the variable x is 2. This distinguishes it from linear equations, where the variable is only raised to the power of 1. The graph of a quadratic equation is a parabola, and depending on the value of the coefficient a, the parabola either opens upwards or downwards.

Interpreting the Equation (4x 2 5x 12 0)

The equation 4x 2 5x 12 0 is written in a simplified format, likely without mathematical operators for brevity or due to formatting limitations. When translated correctly into standard mathematical notation, it becomes:

4x² + 5x – 12 = 0

This is a standard quadratic equation where:

• a = 4

• b = 5

• c = -12

Solving this equation involves finding the values of x that make the equation true, which are known as the roots or solutions of the equation.

Method 1: Solving by Factoring

Factoring is one of the most common and straightforward ways to solve a quadratic equation, but it works best when the equation can be factored easily. In the case of 4x² + 5x – 12 = 0, factoring can be a bit tricky but is still manageable.

To factor, we look for two binomials (expressions in parentheses) such that:

(mx + n)(px + q) = 4x² + 5x – 12

We need to find numbers that multiply to a × c = 4 × -12 = -48 and add to b = 5.

Let’s test factor pairs of -48:

• 8 and -6 work because 8 × -6 = -48 and 8 + (-6) = 2 → too small

• 12 and -4 = 8 → too large

• -3 and 16 = 13 → too large

• -3 and 16 → not working…

Eventually, we find that -3 and 16 are not suitable. Actually, we made a mistake: we need numbers that multiply to -48 and add to 5. That pair is:

-3 and 16

Now, we rewrite the middle term (5x) using -3x and +16x:

4x² – 3x + 16x – 12 = 0

Group terms:

(4x² – 3x) + (16x – 12) = 0

Factor each group:

x(4x – 3) + 4(4x – 3) = 0

Now factor the common binomial:

(x + 4)(4x – 3) = 0

Set each factor equal to zero:

1. x + 4 = 0 → x = -4

2. 4x – 3 = 0 → x = 3/4

So, the solutions to the equation 4x² + 5x – 12 = 0 are:

x = -4 and x = 3/4

Method 2: Solving Using the Quadratic Formula

Another reliable way to solve any quadratic equation, whether it can be factored or not, is by using the quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}x=2a−b±b2−4ac​​

Plugging in the values from our equation 4x² + 5x – 12 = 0 (a = 4, b = 5, c = -12):

x=−5±(5)2−4(4)(−12)2(4)x = \frac{-5 \pm \sqrt{(5)^2 – 4(4)(-12)}}{2(4)}x=2(4)−5±(5)2−4(4)(−12)​​ x=−5±25+1928x = \frac{-5 \pm \sqrt{25 + 192}}{8}x=8−5±25+192​​ x=−5±2178x = \frac{-5 \pm \sqrt{217}}{8}x=8−5±217​​

Since 217 is not a perfect square, we can’t simplify the square root further. The final answers are:

x=−5+2178andx=−5−2178x = \frac{-5 + \sqrt{217}}{8} \quad \text{and} \quad x = \frac{-5 – \sqrt{217}}{8}x=8−5+217​​andx=8−5−217​​

These are the exact solutions, and they approximate to decimal values if needed. This method is especially useful when factoring is not feasible.

Understanding the Discriminant

The discriminant is the part of the quadratic formula under the square root: b² – 4ac. It tells us about the nature of the roots:

• If the discriminant is positive, there are two real and distinct solutions.

• If it is zero, there is one real repeated solution.

• If it is negative, there are two complex (non-real) solutions.

In our equation, the discriminant is:

25 + 192 = 217

Since it is positive, we confirm that the equation 4x² + 5x – 12 = 0 has two real and distinct solutions, just as we found using both factoring and the quadratic formula.

Real-Life Applications of Quadratic Equations

Quadratic equations like 4x² + 5x – 12 = 0 are not just abstract math problems—they have real-world relevance in many fields:

• Physics: To model projectile motion. The height of a ball thrown in the air over time is a quadratic function.

• Engineering: Used in calculations involving structures, optimization problems, and designing curves.

• Economics: To determine profit maximization or cost minimization.

• Biology & Medicine: Growth models and reaction rates sometimes follow quadratic patterns.

Being able to recognize and solve a quadratic equation like 4x 2 5x 12 0 equips you with tools that apply beyond the classroom.

Common Mistakes to Avoid

When solving quadratic equations, especially those like 4x 2 5x 12 0, students often make a few common mistakes:

1. Misreading the Equation Format: Ensure you correctly translate shorthand into proper mathematical format (e.g., 4x² + 5x – 12 = 0).

2. Sign Errors: A small sign error (plus instead of minus) can drastically change your answer.

3. Incorrect Factoring: Not all quadratics factor easily. If you’re stuck, the quadratic formula is a safe fallback.

4. Forgetting to Set the Equation to Zero: You can’t solve a quadratic unless one side is set to zero.

Final Thoughts

Solving the equation 4x 2 5x 12 0 (or properly written as 4x² + 5x – 12 = 0) may look daunting at first, but with an understanding of quadratic principles, it becomes a straightforward task. Whether using factoring, the quadratic formula, or understanding the discriminant, each method gives valuable insight into the equation and its solutions. Beyond academics, mastering quadratics prepares you for complex problem-solving in real life. Keep practicing, stay curious, and remember: every complex-looking problem becomes simpler once you break it down.