Given that $ax^2 + bx + c = 0$, indicate that $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
Understand the Problem
The question presents the quadratic equation $ax^2 + bx + c = 0$ and asks to verify that the solution to this equation, $x$, is correctly given by the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This is a standard problem in algebra.
Answer
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Answer for screen readers
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Steps to Solve
- Start with the given quadratic equation
We begin with the general form of a quadratic equation:
$$ ax^2 + bx + c = 0 $$
- Divide the entire equation by $a$
To simplify, we divide each term by $a$ (assuming $a \ne 0$):
$$ x^2 + \frac{b}{a}x + \frac{c}{a} = 0 $$
- Move the constant term to the right side
Subtract $\frac{c}{a}$ from both sides of the equation:
$$ x^2 + \frac{b}{a}x = -\frac{c}{a} $$
- Complete the square
To complete the square, we need to add $(\frac{1}{2} \cdot \frac{b}{a})^2 = (\frac{b}{2a})^2 = \frac{b^2}{4a^2}$ to both sides of the equation:
$$ x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} = -\frac{c}{a} + \frac{b^2}{4a^2} $$
- Rewrite the left side as a squared term
The left side now forms a perfect square:
$$ (x + \frac{b}{2a})^2 = -\frac{c}{a} + \frac{b^2}{4a^2} $$
- Simplify the right side
Find a common denominator for the right side:
$$ (x + \frac{b}{2a})^2 = \frac{b^2 - 4ac}{4a^2} $$
- Take the square root of both sides
$$ x + \frac{b}{2a} = \pm \sqrt{\frac{b^2 - 4ac}{4a^2}} $$
- Simplify the square root
$$ x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a} $$
- Isolate $x$
Subtract $\frac{b}{2a}$ from both sides to solve for $x$:
$$ x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a} $$
- Combine the terms
Combine the fractions since they have a common denominator:
$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
More Information
The quadratic formula is a fundamental tool for solving quadratic equations. The expression inside the square root, $b^2 - 4ac$, is called the discriminant. The discriminant tells us about the nature of the solutions:
- If $b^2 - 4ac > 0$, there are two distinct real solutions.
- If $b^2 - 4ac = 0$, there is exactly one real solution (a repeated root).
- If $b^2 - 4ac < 0$, there are two complex solutions.
Tips
A common mistake is to incorrectly apply the order of operations or to make errors when simplifying the square root and combining terms in the quadratic formula. Another mistake is sign errors, especially when moving terms from one side of the equation to the other.
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