By completing the square, bnow that the solutions to
are given by

Ask by Harmon Wheeler. in South Africa

Mar 25,2025

Question

By completing the square, bnow that the solutions to $$ a x^{a}+b x+c=0 $$ are given by $$ x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} $$

UpStudy AI · Accepted Answer

To complete the square for the quadratic equation $$ ax^2 + bx + c = 0 $$, we will follow these steps:

1. **Divide by $$ a $$** (if $$ a \neq 0 $$):
   $$
   x^2 + \frac{b}{a}x + \frac{c}{a} = 0
   $$

2. **Rearrange the equation**:
   $$
   x^2 + \frac{b}{a}x = -\frac{c}{a}
   $$

3. **Complete the square**:
   - Take half of the coefficient of $$ x $$ (which is $$ \frac{b}{a} $$), square it, and add it to both sides:
   $$
   x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2
   $$

4. **Rewrite the left side as a square**:
   $$
   \left(x + \frac{b}{2a}\right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2}
   $$

5. **Combine the right side**:
   $$
   \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}
   $$

6. **Take the square root of both sides**:
   $$
   x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a}
   $$

7. **Isolate $$ x $$**:
   $$
   x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a}
   $$

8. **Combine the terms**:
   $$
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   $$

Thus, we have derived the quadratic formula $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ by completing the square.
To solve the quadratic equation $$ ax^2 + bx + c = 0 $$ by completing the square, follow these steps: 1. Divide both sides by $$ a $$ (if $$ a \neq 0 $$). 2. Move the constant term to the other side. 3. Add the square of half the coefficient of $$ x $$ to both sides. 4. Write the left side as a perfect square. 5. Take the square root of both sides. 6. Solve for $$ x $$. The solutions are: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

By completing the square, bnow that the solutions to
are given by

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By completing the square, bnow that the solutions to are given by