Use the quadratic formula to solve for $$ x $$, rounding your answers to 2 decimal places: $$ \begin{array}{lll} ext { a } 3 x^{2}+5 x-1=0 & ext { b }-2 x^{2}+x+7=0 & ext { c } 5 x^{2}-8 x+1=0 \ ext { d } 7 x^{2}=5 x+1 & ext { e } 3 x^{2}+2 x=2 & ext { f } 25 x^{2}+1=20 x\end{array} $$

Question

Use the quadratic formula to solve for $$ x $$, rounding your answers to 2 decimal places: $$ \begin{array}{lll}	ext { a } 3 x^{2}+5 x-1=0 & 	ext { b }-2 x^{2}+x+7=0 & 	ext { c } 5 x^{2}-8 x+1=0 \ 	ext { d } 7 x^{2}=5 x+1 & 	ext { e } 3 x^{2}+2 x=2 & 	ext { f } 25 x^{2}+1=20 x\end{array} $$

UpStudy AI · Accepted Answer

To solve the quadratic equations using the quadratic formula, we will first rewrite each equation in the standard form $$ ax^2 + bx + c = 0 $$ and then apply the quadratic formula:

$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$

Let's rewrite the equations:

1. $$ 3x^2 + 5x - 1 = 0 $$ (a)
2. $$ -2x^2 + x + 7 = 0 $$ (b) → $$ 2x^2 - x - 7 = 0 $$
3. $$ 5x^2 - 8x + 1 = 0 $$ (c)
4. $$ 7x^2 - 5x - 1 = 0 $$ (d)
5. $$ 3x^2 + 2x - 2 = 0 $$ (e)
6. $$ 25x^2 - 20x + 1 = 0 $$ (f)

Now, I will use the quadratic formula for each equation. Let's compute the solutions for all six equations simultaneously.
Solve the quadratic equation by following steps:
- step0: Solve using the quadratic formula:
  $$5x^{2}-8x+1=0$$
- step1: Solve using the quadratic formula:
  $$x=\frac{8\pm \sqrt{\left(-8\right)^{2}-4\times 5}}{2\times 5}$$
- step2: Simplify the expression:
  $$x=\frac{8\pm \sqrt{\left(-8\right)^{2}-4\times 5}}{10}$$
- step3: Simplify the expression:
  $$x=\frac{8\pm \sqrt{44}}{10}$$
- step4: Simplify the expression:
  $$x=\frac{8\pm 2\sqrt{11}}{10}$$
- step5: Separate into possible cases:
  $$\begin{align}&x=\frac{8+2\sqrt{11}}{10}\&x=\frac{8-2\sqrt{11}}{10}\end{align}$$
- step6: Simplify the expression:
  $$\begin{align}&x=\frac{4+\sqrt{11}}{5}\&x=\frac{8-2\sqrt{11}}{10}\end{align}$$
- step7: Simplify the expression:
  $$\begin{align}&x=\frac{4+\sqrt{11}}{5}\&x=\frac{4-\sqrt{11}}{5}\end{align}$$
- step8: Rewrite:
  $$x_{1}=\frac{4-\sqrt{11}}{5},x_{2}=\frac{4+\sqrt{11}}{5}$$
Solve the equation $$ 3x^{2}+5x-1=0 $$.
Solve the quadratic equation by following steps:
- step0: Solve using the quadratic formula:
  $$3x^{2}+5x-1=0$$
- step1: Solve using the quadratic formula:
  $$x=\frac{-5\pm \sqrt{5^{2}-4\times 3\left(-1\right)}}{2\times 3}$$
- step2: Simplify the expression:
  $$x=\frac{-5\pm \sqrt{5^{2}-4\times 3\left(-1\right)}}{6}$$
- step3: Simplify the expression:
  $$x=\frac{-5\pm \sqrt{37}}{6}$$
- step4: Separate into possible cases:
  $$\begin{align}&x=\frac{-5+\sqrt{37}}{6}\&x=\frac{-5-\sqrt{37}}{6}\end{align}$$
- step5: Rewrite the fraction:
  $$\begin{align}&x=\frac{-5+\sqrt{37}}{6}\&x=-\frac{5+\sqrt{37}}{6}\end{align}$$
- step6: Rewrite:
  $$x_{1}=-\frac{5+\sqrt{37}}{6},x_{2}=\frac{-5+\sqrt{37}}{6}$$
Solve the equation $$ 25x^{2}-20x+1=0 $$.
Solve the quadratic equation by following steps:
- step0: Solve using the quadratic formula:
  $$25x^{2}-20x+1=0$$
- step1: Solve using the quadratic formula:
  $$x=\frac{20\pm \sqrt{\left(-20\right)^{2}-4\times 25}}{2\times 25}$$
- step2: Simplify the expression:
  $$x=\frac{20\pm \sqrt{\left(-20\right)^{2}-4\times 25}}{50}$$
- step3: Simplify the expression:
  $$x=\frac{20\pm \sqrt{300}}{50}$$
- step4: Simplify the expression:
  $$x=\frac{20\pm 10\sqrt{3}}{50}$$
- step5: Separate into possible cases:
  $$\begin{align}&x=\frac{20+10\sqrt{3}}{50}\&x=\frac{20-10\sqrt{3}}{50}\end{align}$$
- step6: Simplify the expression:
  $$\begin{align}&x=\frac{2+\sqrt{3}}{5}\&x=\frac{20-10\sqrt{3}}{50}\end{align}$$
- step7: Simplify the expression:
  $$\begin{align}&x=\frac{2+\sqrt{3}}{5}\&x=\frac{2-\sqrt{3}}{5}\end{align}$$
- step8: Rewrite:
  $$x_{1}=\frac{2-\sqrt{3}}{5},x_{2}=\frac{2+\sqrt{3}}{5}$$
Solve the equation $$ 2x^{2}-x-7=0 $$.
Solve the quadratic equation by following steps:
- step0: Solve using the quadratic formula:
  $$2x^{2}-x-7=0$$
- step1: Solve using the quadratic formula:
  $$x=\frac{1\pm \sqrt{\left(-1\right)^{2}-4\times 2\left(-7\right)}}{2\times 2}$$
- step2: Simplify the expression:
  $$x=\frac{1\pm \sqrt{\left(-1\right)^{2}-4\times 2\left(-7\right)}}{4}$$
- step3: Simplify the expression:
  $$x=\frac{1\pm \sqrt{57}}{4}$$
- step4: Separate into possible cases:
  $$\begin{align}&x=\frac{1+\sqrt{57}}{4}\&x=\frac{1-\sqrt{57}}{4}\end{align}$$
- step5: Rewrite:
  $$x_{1}=\frac{1-\sqrt{57}}{4},x_{2}=\frac{1+\sqrt{57}}{4}$$
Solve the equation $$ 7x^{2}-5x-1=0 $$.
Solve the quadratic equation by following steps:
- step0: Solve using the quadratic formula:
  $$7x^{2}-5x-1=0$$
- step1: Solve using the quadratic formula:
  $$x=\frac{5\pm \sqrt{\left(-5\right)^{2}-4\times 7\left(-1\right)}}{2\times 7}$$
- step2: Simplify the expression:
  $$x=\frac{5\pm \sqrt{\left(-5\right)^{2}-4\times 7\left(-1\right)}}{14}$$
- step3: Simplify the expression:
  $$x=\frac{5\pm \sqrt{53}}{14}$$
- step4: Separate into possible cases:
  $$\begin{align}&x=\frac{5+\sqrt{53}}{14}\&x=\frac{5-\sqrt{53}}{14}\end{align}$$
- step5: Rewrite:
  $$x_{1}=\frac{5-\sqrt{53}}{14},x_{2}=\frac{5+\sqrt{53}}{14}$$
Solve the equation $$ 3x^{2}+2x-2=0 $$.
Solve the quadratic equation by following steps:
- step0: Solve using the quadratic formula:
  $$3x^{2}+2x-2=0$$
- step1: Solve using the quadratic formula:
  $$x=\frac{-2\pm \sqrt{2^{2}-4\times 3\left(-2\right)}}{2\times 3}$$
- step2: Simplify the expression:
  $$x=\frac{-2\pm \sqrt{2^{2}-4\times 3\left(-2\right)}}{6}$$
- step3: Simplify the expression:
  $$x=\frac{-2\pm \sqrt{28}}{6}$$
- step4: Simplify the expression:
  $$x=\frac{-2\pm 2\sqrt{7}}{6}$$
- step5: Separate into possible cases:
  $$\begin{align}&x=\frac{-2+2\sqrt{7}}{6}\&x=\frac{-2-2\sqrt{7}}{6}\end{align}$$
- step6: Simplify the expression:
  $$\begin{align}&x=\frac{-1+\sqrt{7}}{3}\&x=\frac{-2-2\sqrt{7}}{6}\end{align}$$
- step7: Simplify the expression:
  $$\begin{align}&x=\frac{-1+\sqrt{7}}{3}\&x=-\frac{1+\sqrt{7}}{3}\end{align}$$
- step8: Rewrite:
  $$x_{1}=-\frac{1+\sqrt{7}}{3},x_{2}=\frac{-1+\sqrt{7}}{3}$$
Here are the solutions for each quadratic equation, rounded to two decimal places:

1. For $$ 3x^2 + 5x - 1 = 0 $$:
   - $$ x_1 \approx -1.85 $$
   - $$ x_2 \approx 0.18 $$

2. For $$ -2x^2 + x + 7 = 0 $$ (rewritten as $$ 2x^2 - x - 7 = 0 $$):
   - $$ x_1 \approx -1.64 $$
   - $$ x_2 \approx 2.14 $$

3. For $$ 5x^2 - 8x + 1 = 0 $$:
   - $$ x_1 \approx 0.14 $$
   - $$ x_2 \approx 1.46 $$

4. For $$ 7x^2 - 5x - 1 = 0 $$:
   - $$ x_1 \approx -0.16 $$
   - $$ x_2 \approx 0.88 $$

5. For $$ 3x^2 + 2x - 2 = 0 $$:
   - $$ x_1 \approx -1.22 $$
   - $$ x_2 \approx 0.55 $$

6. For $$ 25x^2 - 20x + 1 = 0 $$:
   - $$ x_1 \approx 0.05 $$
   - $$ x_2 \approx 0.75 $$

These are the rounded solutions for each equation.
Here are the solutions for each quadratic equation, rounded to two decimal places: 1. $$ 3x^2 + 5x - 1 = 0 $$: - $$ x \approx -1.85 $$ - $$ x \approx 0.18 $$ 2. $$ -2x^2 + x + 7 = 0 $$ (rewritten as $$ 2x^2 - x - 7 = 0 $$): - $$ x \approx -1.64 $$ - $$ x \approx 2.14 $$ 3. $$ 5x^2 - 8x + 1 = 0 $$: - $$ x \approx 0.14 $$ - $$ x \approx 1.46 $$ 4. $$ 7x^2 - 5x - 1 = 0 $$: - $$ x \approx -0.16 $$ - $$ x \approx 0.88 $$ 5. $$ 3x^2 + 2x - 2 = 0 $$: - $$ x \approx -1.22 $$ - $$ x \approx 0.55 $$ 6. $$ 25x^2 - 20x + 1 = 0 $$: - $$ x \approx 0.05 $$ - $$ x \approx 0.75 $$

Use the quadratic formula to solve for , rounding your answers to 2 decimal places:

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