How to Solve the Equation x² - x = 144 Step by Step
Quick Answer
To solve the equation x² - x = 144, first rearrange it to x² - x - 144 = 0. Then, you can use factoring or the quadratic formula to find the values of x.
To solve the equation $$x^2 - x = 144$$, we need to rearrange it to a standard form of a quadratic equation. The first step is to move all terms to one side of the equation. This means subtracting 144 from both sides:
$$x^2 - x - 144 = 0$$
Now, we have a quadratic equation of the form $$ax^2 + bx + c = 0$$ where:
- $$a = 1$$ (the coefficient of $$x^2$$)
- $$b = -1$$ (the coefficient of $$x$$)
- $$c = -144$$ (the constant term)
### Understanding Quadratic Equations
Quadratic equations are important in algebra, and they can represent various real-world situations, such as projectile motion or profit maximization. The general solution for a quadratic equation can be found using factoring or the quadratic formula:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
### Solving by Factoring
To solve $$x^2 - x - 144 = 0$$ by factoring, we need to find two numbers that multiply to -144 (the constant) and add to -1 (the coefficient of $$x$$). The numbers that work here are 12 and -12.
So we can rewrite the equation as:
$$(x - 12)(x + 12) = 0$$
Setting each factor to zero gives us:
- $$x - 12 = 0$$ → $$x = 12$$
- $$x + 12 = 0$$ → $$x = -12$$
### Solving Using the Quadratic Formula
If factoring seems difficult or if the equation does not factor nicely, you can always use the quadratic formula. Plugging in our values:
- $$a = 1$$, $$b = -1$$, $$c = -144$$
We calculate:
1. $$b^2 - 4ac = (-1)^2 - 4(1)(-144) = 1 + 576 = 577$$
2. Now, apply the quadratic formula:
$$x = \frac{-(-1) \pm \sqrt{577}}{2(1)}$$
3. Simplifying this gives us:
$$x = \frac{1 \pm \sqrt{577}}{2}$$
This means that the equation can have two solutions: approximately 12.29 and -11.29.
### Conclusion
In summary, to solve $$x^2 - x = 144$$, you first rearrange the equation to standard form and then either factor or use the quadratic formula to find the solutions for $$x$$. Understanding these steps will help you tackle more complex equations in the future. Remember, practice is key to mastering quadratic equations!
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