QuestionSolve: $\left(2 a^{2}+5 a+2\right)^{\frac{1}{2}}=3$ $a=-\frac{7}{2}$ $a=1$ $a=-\frac{7}{2}$ or $a=1$ no real solution DONE -

Studdy Solution

STEP 1

What is this asking? We're trying to find the value(s) of $a$ that make the square root of $2a^2 + 5a + 2$ equal to 3! Watch out! Remember, squaring both sides of an equation can sometimes introduce extra solutions that don't actually work, so we must check our answers at the end!

STEP 2

1. Eliminate the Square Root
2. Set Up a Quadratic Equation
3. Solve for a
4. Check Our Solutions

STEP 3

To get rid of that pesky square root, let's square both sides of the equation!
This gives us: $(2a^2 + 5a + 2)^{\frac{1}{2}} \cdot (2a^2 + 5a + 2)^{\frac{1}{2}} = 3 \cdot 3$ $2a^2 + 5a + 2 = 9$ Now, that looks much more manageable!

STEP 4

Let's rearrange our equation to get it into standard quadratic form, which is $ax^2 + bx + c = 0$ .
We want zero on one side, so we'll subtract 9 from both sides: $2a^2 + 5a + 2 - 9 = 9 - 9$ $2a^2 + 5a - 7 = 0$ Perfect! Now we have a nice, clean quadratic equation ready to be solved.

STEP 5

Let's see if we can factor this quadratic.
We're looking for two numbers that multiply to $(2) \cdot (-7) = -14$ and add up to 5.
Those numbers are 7 and -2!
So, we rewrite the middle term and factor by grouping: $2a^2 - 2a + 7a - 7 = 0$ $2a(a - 1) + 7(a - 1) = 0$ $(2a + 7)(a - 1) = 0$

STEP 6

Now, we set each factor equal to zero and solve for $a$ : $2a + 7 = 0 \implies 2a = -7 \implies a = -\frac{7}{2}$ $a - 1 = 0 \implies a = 1$ We have two possible solutions: $a = -\frac{7}{2}$ and $a = 1$ .
But wait!
We squared both sides of the equation earlier, so we need to check if these solutions actually work!

STEP 7

Let's plug $a = -\frac{7}{2}$ back into the original equation: $\sqrt{2(-\frac{7}{2})^2 + 5(-\frac{7}{2}) + 2} = \sqrt{2(\frac{49}{4}) - \frac{35}{2} + 2} = \sqrt{\frac{49}{2} - \frac{35}{2} + \frac{4}{2}} = \sqrt{\frac{18}{2}} = \sqrt{9} = 3$ It works!

STEP 8

Now let's check $a = 1$ : $\sqrt{2(1)^2 + 5(1) + 2} = \sqrt{2 + 5 + 2} = \sqrt{9} = 3$ This one works too!

STEP 9

Both solutions check out!
So, the solutions are $a = -\frac{7}{2}$ and $a = 1$ .

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