Introduction
Ever stumbled upon an equation like 4x 2 5x 12 0 and felt stuck? You’re not alone! Quadratic equations are a crucial part of algebra, and solving them can be easier than you think.
In this guide, we’ll walk you through three common methods to solve the equation 4x 2 5x 12 0 step by step. By the end, you’ll feel confident tackling any quadratic equation that comes your way.
Let’s dive in!
What is a Quadratic Equation?
A quadratic equation is any equation that takes the form: ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0
Where:
- a, b, and c are constants
- x is the variable
- a ≠ 0 (otherwise, it’s not quadratic!)
For our specific equation: 4×2+5x−12=04x² + 5x – 12 = 04×2+5x−12=0
- a = 4
- b = 5
- c = -12
Method 1: Factoring
Factoring is the simplest way to solve a quadratic equation if it can be factored. Let’s see if we can factor 4x² + 5x – 12 = 0.
Step 1: Multiply ‘a’ and ‘c’
First, multiply the coefficient of x² (4) and the constant term (-12): 4×(−12)=−484 \times (-12) = -484×(−12)=−48
Now, find two numbers that multiply to -48 and add up to 5 (the coefficient of x).
The numbers +8 and -6 work because: 8×(−6)=−488 \times (-6) = -488×(−6)=−48 8+(−6)=2(not correct!)8 + (-6) = 2 \quad (\text{not correct!})8+(−6)=2(not correct!)
Oops! No two numbers satisfy this condition, meaning this equation cannot be factored easily. So, let’s move on to the quadratic formula.
Method 2: Using the Quadratic Formula
When factoring doesn’t work, the quadratic formula is your best friend: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}x=2a−b±b2−4ac
Step 1: Plug in the values
We have:
- 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)
Step 2: Solve under the square root
First, calculate the discriminant (the part under the square root): 52−4(4)(−12)=25+192=2175^2 – 4(4)(-12) = 25 + 192 = 21752−4(4)(−12)=25+192=217
Since 217 is not a perfect square, we keep it as √217.
Step 3: Find the two values of x
x=−5±2178x = \frac{-5 \pm \sqrt{217}}{8}x=8−5±217
Approximating √217 ≈ 14.73, we get: x=−5+14.738=9.738≈1.22x = \frac{-5 + 14.73}{8} = \frac{9.73}{8} \approx 1.22x=8−5+14.73=89.73≈1.22 x=−5−14.738=−19.738≈−2.47x = \frac{-5 – 14.73}{8} = \frac{-19.73}{8} \approx -2.47x=8−5−14.73=8−19.73≈−2.47
Final Answer:
x≈1.22,−2.47x \approx 1.22, -2.47x≈1.22,−2.47
Method 3: Completing the Square
Completing the square is another method, but it can be tedious when dealing with a leading coefficient other than 1. For simplicity, we’ll skip this method here, but feel free to explore it!
Conclusion
We successfully solved the quadratic equation 4x² + 5x – 12 = 0 using the quadratic formula since factoring didn’t work.
Final Answer:
x≈1.22,−2.47x \approx 1.22, -2.47x≈1.22,−2.47
Now, the next time you see a quadratic equation, you know exactly how to handle it!
FAQs
1. What is the fastest way to solve a quadratic equation?
If the equation can be factored, that’s the fastest way. Otherwise, the quadratic formula works for all cases.
2. What if the discriminant is negative?
If b² – 4ac < 0, then the equation has no real solutions, only complex (imaginary) ones.
3. Can I use a calculator to solve quadratic equations?
Yes! Many scientific calculators have a quadratic solver feature.
4. Why does completing the square take longer?
It requires more steps, especially when a ≠ 1, but it’s useful for learning how quadratics work.
