Solve the following quadratic equation using square roots: −9+(x−3)2/−4=−4
To solve the quadratic equation using square roots, we will first simplify the equation:
-9 + (x - 3)^2 / -4 = -4
Multiply both sides of the equation by -4 to eliminate the fraction:
4(-9) + (x - 3)^2 = 16
-36 + (x - 3)^2 = 16
Next, move -36 to the right side of the equation:
(x - 3)^2 = 16 + 36
(x - 3)^2 = 52
Now, take the square root of both sides of the equation:
√(x - 3)^2 = ±√52
x - 3 = ±√52
Now, solve for x by adding 3 to both sides of the equation:
x = 3 ± √52
Therefore, the solution to the quadratic equation is:
x = 3 + √52 or x = 3 - √52
To solve the quadratic equation −9+(x−3)2/−4=−4 using square roots, we need to isolate the variable, x, and then solve for it.
Here are the steps to follow:
Step 1: Remove the fraction
First, we need to get rid of the fraction by multiplying both sides of the equation by -4. This gives us:
-4 * (-9 + (x - 3)² / -4) = -4 * -4
Simplifying the equation gives us:
36 - (x - 3)² = 16
Step 2: Expand the square
Expand the square on the left side of the equation:
36 - (x - 3)(x - 3) = 16
Expanding the square gives us:
36 - (x² - 6x + 9) = 16
Simplifying the equation further yields:
36 - x² + 6x - 9 = 16
Step 3: Rearrange and combine like terms
Rearrange the terms so that the equation is in standard quadratic form (ax² + bx + c = 0):
-x² + 6x + 27 = 16
Combine like terms:
-x² + 6x + 27 - 16 = 0
Simplifying the equation further gives us:
-x² + 6x + 11 = 0
Step 4: Solve for x using the quadratic formula
To solve the quadratic equation -x² + 6x + 11 = 0, we can use the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
In our case, a = -1, b = 6, and c = 11. Substituting these values into the quadratic formula:
x = (-(6) ± √((6)² - 4(-1)(11))) / (2(-1))
Simplifying the expression further gives us:
x = (-6 ± √(36 + 44)) / (-2)
x = (-6 ± √80) / (-2)
Step 5: Simplify the square root
Simplify the square root of 80:
x = (-6 ± √(16 * 5)) / (-2)
x = (-6 ± 4√5) / (-2)
Step 6: Simplify further
Simplify the expression by dividing both the numerator and denominator by -2:
x = -6/(-2) ± 4√5/(-2)
x = 3 ± 2√5
Therefore, the solutions to the quadratic equation are x = 3 + 2√5 and x = 3 - 2√5.
To solve the given quadratic equation using square roots, we need to isolate the squared term, apply the square root to both sides of the equation, and solve for x.
Let's break down the steps to solve the quadratic equation:
Step 1: Distribute the (-4) to the terms inside the parentheses:
-9 + (x - 3)²/(-4) = -4
This simplifies to:
-9 - (x - 3)²/4 = -4
Step 2: Move the constant term to the right side of the equation:
-9 = -4 + (x - 3)²/4
Step 3: Add 4 to both sides of the equation to eliminate the constant term on the right side:
-9 + 4 = (x - 3)²/4
This simplifies to:
-5 = (x - 3)²/4
Step 4: Multiply both sides of the equation by 4 to eliminate the fraction:
-5 * 4 = (x - 3)²
This becomes:
-20 = (x - 3)²
Step 5: Take the square root of both sides to solve for x:
√(-20) = √[(x - 3)²]
Step 6: Simplify the left side:
Since the square root of a negative number is imaginary, we cannot proceed further. Therefore, this quadratic equation has no solutions in the real number system.
In conclusion, the given quadratic equation has no solutions in the real number system.