Question 4 (1 point) solve. $$ 3(x+7)^{2}+17=59 $$ a $$ 42 \pm \sqrt{14} $$ b $$ -7 \pm \sqrt{14} $$ c $$ -7 \pm \sqrt{13} $$ d $$ 42 \pm \sqrt{13} $$

Question

UpStudy AI · Accepted Answer

Solve the quadratic equation by following steps:
- step0: Solve using the quadratic formula:
  $$3\left(x+7\right)^{2}+17=59$$
- step1: Expand the expression:
  $$3x^{2}+42x+164=59$$
- step2: Move the expression to the left side:
  $$3x^{2}+42x+105=0$$
- step3: Solve using the quadratic formula:
  $$x=\frac{-42\pm \sqrt{42^{2}-4\times 3\times 105}}{2\times 3}$$
- step4: Simplify the expression:
  $$x=\frac{-42\pm \sqrt{42^{2}-4\times 3\times 105}}{6}$$
- step5: Simplify the expression:
  $$x=\frac{-42\pm \sqrt{504}}{6}$$
- step6: Simplify the expression:
  $$x=\frac{-42\pm 6\sqrt{14}}{6}$$
- step7: Separate into possible cases:
  $$\begin{align}&x=\frac{-42+6\sqrt{14}}{6}\&x=\frac{-42-6\sqrt{14}}{6}\end{align}$$
- step8: Simplify the expression:
  $$\begin{align}&x=-7+\sqrt{14}\&x=\frac{-42-6\sqrt{14}}{6}\end{align}$$
- step9: Simplify the expression:
  $$\begin{align}&x=-7+\sqrt{14}\&x=-7-\sqrt{14}\end{align}$$
- step10: Rewrite:
  $$x_{1}=-7-\sqrt{14},x_{2}=-7+\sqrt{14}$$
To solve the equation

$$
3(x+7)^{2}+17=59,
$$

we can follow these steps:

1. **Isolate the quadratic term**: Start by subtracting 17 from both sides:

$$
   3(x+7)^{2} = 59 - 17
   $$

This simplifies to:

$$
   3(x+7)^{2} = 42.
   $$

2. **Divide by 3**: Next, divide both sides by 3:

$$
   (x+7)^{2} = \frac{42}{3} = 14.
   $$

3. **Take the square root**: Now, take the square root of both sides:

$$
   x+7 = \pm \sqrt{14}.
   $$

4. **Solve for $$x$$**: Finally, subtract 7 from both sides:

$$
   x = -7 \pm \sqrt{14}.
   $$

Thus, the solutions are

$$
x_{1} = -7 - \sqrt{14}, \quad x_{2} = -7 + \sqrt{14}.
$$

The correct answer corresponds to option **b**: $$ -7 \pm \sqrt{14} $$.
The solutions are $$ x = -7 \pm \sqrt{14} $$, so the correct answer is option b.

Question 4 (1 point) solve. \[ 3(x+7)^{2}+17=59 \] a \( 42 \pm \sqrt{14} \) b \( -7 \pm \sqrt{14} \) c \( -7 \pm \sqrt{13} \) d \( 42 \pm \sqrt{13} \)

Solución de inteligencia artificial de Upstudy

Responder

Solución

Bonus Knowledge

preguntas relacionadas

Latest Algebra Questions