how to prove the quadratic formula|quadratic equations |
The video explains quadratic equations, highlighting how they are used to solve problems involving variables raised to the power of two. It covers topics such as the standard form of a quadratic equation, methods for solving quadratic equations (including factoring, completing the square, and using the quadratic formula), and real-life applications of quadratic equations.
To prove the quadratic formula, we start with a general quadratic equation of the form:
\[ ax^2 + bx + c = 0 \]
Step 1: Divide each term by \(a\) to simplify the equation:
\[ x^2 + \frac{b}{a}x + \frac{c}{a} = 0 \]
Step 2: Complete the square by adding and subtracting \(\left(\frac{b}{2a}\right)^2\) inside the parentheses of the \(x\) terms:
\[ x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 + \frac{c}{a} = 0 \]
Step 3: Factor the trinomial inside the parentheses and simplify:
\[ \left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 + \frac{c}{a} = 0 \]
Step 4: Expand the squared term:
\[ \left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a^2} + \frac{c}{a} = 0 \]
Step 5: Rearrange the terms:
\[ \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} \]
Step 6: Take the square root of both sides:
\[ x + \frac{b}{2a} = \pm \sqrt{\frac{b^2 - 4ac}{4a^2}} \]
Step 7: Simplify the square root:
\[ x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a} \]
Step 8: Solve for \(x\):
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This is the quadratic formula that we set out to prove. It provides the solutions to a general quadratic equation of the form \(ax^2 + bx + c = 0\).
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