Find the solution(s) using the quadratic formula. 4 ( x − 2 ) 2 − 5 = x + 7 4\left(x-2\right)^2-5=x+7

Here, we want to solve the quadratic four times x minus two squared minus five equals x plus seven using the quadratic formula. Now first, we've got a couple steps before we can actually apply our quadratic formula. We need to get this in standard form. So in order to do that, we need to foil out all of our parentheses and then combine all of our like terms. So let's start with x minus two squared. I can rewrite that as x minus two times x minus two. And then using our foil technique, we'd have x times x is x squared, x times negative two, negative two x, Negative two times x, another negative two x. And negative two times negative two makes positive four. So combining like terms, we get x squared minus four x plus four. So now I'm going to take this and plug it back into my original equation, and we'll have four times x squared minus four x plus four minus five equals x plus seven. Next, let's distribute in this four into this quadratic. Doing that, we get four x squared minus 16 x plus 16 minus five equals x plus seven. Now we're going to combine some like terms of 16 and negative five. So we have four x squared minus 16 x plus 11 equals x plus seven. And now let's subtract x and seven over to the other side of our equation and combine like terms. So I'll get four x squared, negative 16 x minus x will give us negative 17 x, so minus 17 x. And positive 11 minus seven will combine to make positive four, so we'll have plus four. And that is now all equal to zero. So now that we have a quadratic in standard form, we can apply our quadratic formula. So let's identify our a value as four, our b value as negative 17, and our c value as also four. And plug these in. So we'll get x is equal to negative b, so negative 17, plus or minus square root of b squared, so negative 17 squared, minus four times a, which is four, times c, which is four, all divided by two times a, which is four. Simplifying, we'll have x is equal to negative negative 17, so positive 17, plus or minus negative 17 squared. Using our calculator, we can find is 289 minus four times four times four, also using our calculator, gets us 64. Now we'll all be divided by two times four, which is eight. Now 289 minus 64 is two twenty five, which is a perfectly square number. So this will reduce to x is equal to 17 plus or minus the square root of two twenty five, which is 15, all divided by eight. And now we have our two solutions. We have x is equal to 17 plus 15, which is 32 divided by eight. And that simplifies to 32 divided by eight, which is four. And our second solution is 17 minus 15. So we have x equals two divided by eight, and that simplifies to x equals one fourth. So these are the two solutions to this quadratic equation. We could always check our work by taking these solutions and plugging them back in to make sure the equation is true.

12. Quadratic Equations and Functions

The Quadratic Formula

Beginning & Intermediate Algebra

12. Quadratic Equations and Functions

The Quadratic Formula

Struggling with Beginning & Intermediate Algebra?

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Multiple Choice

Find the solution(s) using the quadratic formula.
$4 {(x - 2)}^{2} - 5 = x + 7$

$x = 5, x = \frac{1}{2}$

$x = 5, x = \frac{1}{4}$

$x = 4, x = \frac{1}{4}$

$x = 4, x = \frac{1}{2}$

Verified step by step guidance

Start with the given equation: \$4\left(x - 2\right)^2 - 5 = x + 7$.

First, expand the squared term: $\left(x - 2\right)^2 = x^2 - 4x + 4$. Substitute this back into the equation to get \$4(x^2 - 4x + 4) - 5 = x + 7$.

Distribute the 4 across the terms inside the parentheses: \$4x^2 - 16x + 16 - 5 = x + 7$.

Simplify the left side by combining like terms: \$4x^2 - 16x + 11 = x + 7$. Then, move all terms to one side to set the equation equal to zero: $4x^2 - 16x + 11 - x - 7 = 0$, which simplifies to $4x^2 - 17x + 4 = 0$.

Now, apply the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a=4$, $b=-17$, and $c=4$. Substitute these values into the formula to find the solutions for $x$.

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Find the solution(s) using the quadratic formula.
$4 {(x - 2)}^{2} - 5 = x + 7$