Exercises $$ \begin{array}{ll} ext { What are the solutions for each equation? Use the Quadratic Formula. } \ \begin{array}{ll} ext { 1. }-x^{2}+7 x-3=0 & ext { 2. } x^{2}+6 x=10 \ ext { 3. } 2 x^{2}=4 x+3 & ext { 4. } 4 x^{2}+81=36 x\end{array} \ ext { Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. }\end{array} $$

Question

Exercises $$ \begin{array}{ll}	ext { What are the solutions for each equation? Use the Quadratic Formula. } \ \begin{array}{ll}	ext { 1. }-x^{2}+7 x-3=0 & 	ext { 2. } x^{2}+6 x=10 \ 	ext { 3. } 2 x^{2}=4 x+3 & 	ext { 4. } 4 x^{2}+81=36 x\end{array} \ 	ext { Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. }\end{array} $$

UpStudy AI · Accepted Answer

Solve the quadratic equation by following steps:
- step0: Solve by factoring:
  $$4x^{2}-36x+81=0$$
- step1: Factor the expression:
  $$\left(2x-9\right)^{2}=0$$
- step2: Simplify the expression:
  $$2x-9=0$$
- step3: Move the constant to the right side:
  $$2x=0+9$$
- step4: Remove 0:
  $$2x=9$$
- step5: Divide both sides:
  $$\frac{2x}{2}=\frac{9}{2}$$
- step6: Divide the numbers:
  $$x=\frac{9}{2}$$
Solve the equation $$ -x^{2}+7 x-3=0 $$.
Solve the quadratic equation by following steps:
- step0: Solve using the quadratic formula:
  $$-x^{2}+7x-3=0$$
- step1: Multiply both sides:
  $$x^{2}-7x+3=0$$
- step2: Solve using the quadratic formula:
  $$x=\frac{7\pm \sqrt{\left(-7\right)^{2}-4\times 3}}{2}$$
- step3: Simplify the expression:
  $$x=\frac{7\pm \sqrt{37}}{2}$$
- step4: Separate into possible cases:
  $$\begin{align}&x=\frac{7+\sqrt{37}}{2}\&x=\frac{7-\sqrt{37}}{2}\end{align}$$
- step5: Rewrite:
  $$x_{1}=\frac{7-\sqrt{37}}{2},x_{2}=\frac{7+\sqrt{37}}{2}$$
Solve the equation $$ x^{2}+6 x-10=0 $$.
Solve the quadratic equation by following steps:
- step0: Solve using the quadratic formula:
  $$x^{2}+6x-10=0$$
- step1: Solve using the quadratic formula:
  $$x=\frac{-6\pm \sqrt{6^{2}-4\left(-10\right)}}{2}$$
- step2: Simplify the expression:
  $$x=\frac{-6\pm \sqrt{76}}{2}$$
- step3: Simplify the expression:
  $$x=\frac{-6\pm 2\sqrt{19}}{2}$$
- step4: Separate into possible cases:
  $$\begin{align}&x=\frac{-6+2\sqrt{19}}{2}\&x=\frac{-6-2\sqrt{19}}{2}\end{align}$$
- step5: Simplify the expression:
  $$\begin{align}&x=-3+\sqrt{19}\&x=\frac{-6-2\sqrt{19}}{2}\end{align}$$
- step6: Simplify the expression:
  $$\begin{align}&x=-3+\sqrt{19}\&x=-3-\sqrt{19}\end{align}$$
- step7: Rewrite:
  $$x_{1}=-3-\sqrt{19},x_{2}=-3+\sqrt{19}$$
Solve the equation $$ 2 x^{2}-4 x-3=0 $$.
Solve the quadratic equation by following steps:
- step0: Solve using the quadratic formula:
  $$2x^{2}-4x-3=0$$
- step1: Solve using the quadratic formula:
  $$x=\frac{4\pm \sqrt{\left(-4\right)^{2}-4\times 2\left(-3\right)}}{2\times 2}$$
- step2: Simplify the expression:
  $$x=\frac{4\pm \sqrt{\left(-4\right)^{2}-4\times 2\left(-3\right)}}{4}$$
- step3: Simplify the expression:
  $$x=\frac{4\pm \sqrt{40}}{4}$$
- step4: Simplify the expression:
  $$x=\frac{4\pm 2\sqrt{10}}{4}$$
- step5: Separate into possible cases:
  $$\begin{align}&x=\frac{4+2\sqrt{10}}{4}\&x=\frac{4-2\sqrt{10}}{4}\end{align}$$
- step6: Simplify the expression:
  $$\begin{align}&x=\frac{2+\sqrt{10}}{2}\&x=\frac{4-2\sqrt{10}}{4}\end{align}$$
- step7: Simplify the expression:
  $$\begin{align}&x=\frac{2+\sqrt{10}}{2}\&x=\frac{2-\sqrt{10}}{2}\end{align}$$
- step8: Rewrite:
  $$x_{1}=\frac{2-\sqrt{10}}{2},x_{2}=\frac{2+\sqrt{10}}{2}$$
Here are the solutions for each equation using the Quadratic Formula:

1. For the equation $$-x^{2}+7x-3=0$$:
   - The solutions are:
     $$
     x_{1}=\frac{7-\sqrt{37}}{2}, \quad x_{2}=\frac{7+\sqrt{37}}{2}
     $$

2. For the equation $$x^{2}+6x=10$$ (which can be rewritten as $$x^{2}+6x-10=0$$):
   - The solutions are:
     $$
     x_{1}=-3-\sqrt{19}, \quad x_{2}=-3+\sqrt{19}
     $$

3. For the equation $$2x^{2}=4x+3$$ (which can be rewritten as $$2x^{2}-4x-3=0$$):
   - The solutions are:
     $$
     x_{1}=\frac{2-\sqrt{10}}{2}, \quad x_{2}=\frac{2+\sqrt{10}}{2}
     $$

4. For the equation $$4x^{2}+81=36x$$ (which can be rewritten as $$4x^{2}-36x+81=0$$):
   - The solutions are:
     $$
     x=\frac{9}{2} \quad \text{or} \quad x=4.5
     $$

These solutions provide the values of $$x$$ for each quadratic equation.
Here are the solutions for each equation using the Quadratic Formula: 1. $$-x^{2}+7x-3=0$$: $$x_{1}=\frac{7-\sqrt{37}}{2}$$, $$x_{2}=\frac{7+\sqrt{37}}{2}$$ 2. $$x^{2}+6x=10$$: $$x_{1}=-3-\sqrt{19}$$, $$x_{2}=-3+\sqrt{19}$$ 3. $$2x^{2}=4x+3$$: $$x_{1}=\frac{2-\sqrt{10}}{2}$$, $$x_{2}=\frac{2+\sqrt{10}}{2}$$ 4. $$4x^{2}+81=36x$$: $$x=4.5$$

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