Solve
square root of 79-5x =6x+4
(Keep your answers in simplified improper fraction form)
√(79-5x) = 6x+4
79-5x = (6x+4)^2 = 36x^2+48x+16
36x^2+53x-63 = 0
(4x+9)(9x-7) = 0
x = -9/4 or 7/9
but, you have to check for spurious roots:
√(79-5(-9/4)) = √(79+45/4) = 9.5
but, 6(-9/4)+4 = -9.5
This is the spurious root. If you square both sides, it becomes true, but not in the original.
√(79-5(7/9)) = 25/3
6(7/9)+4 = 25/3
So, 7/9 is the only solution
To solve the equation, square root of 79-5x = 6x+4, we need to isolate the variable x. Follow the steps below:
Step 1: Start by squaring both sides of the equation. This will eliminate the square root sign.
(√79 - 5x)^2 = (6x + 4)^2
Step 2: Expand both sides of the equation using FOIL (First, Outer, Inner, Last) method.
(79 - 5x)(79 - 5x) = (6x + 4)(6x + 4)
Step 3: Simplify each side of the equation using the distributive property and combining like terms.
79^2 - 2 * 79 * 5x + (5x)^2 = 6x * 6x + 2 * 6x * 4 + 4^2
6241 - 790x + 25x^2 = 36x^2 + 48x + 16
Step 4: Rearrange the equation to set it equal to zero by subtracting both sides by their respective terms.
0 = 36x^2 + 48x + 16 - 25x^2 - 790x + 6241
Simplifying further, we get:
0 = 11x^2 - 742x + 6225
Step 5: To solve this quadratic equation, we can either use factoring, completing the square method, or the quadratic formula. In this case, the quadratic equation does not factor easily, so we will use the quadratic formula.
The quadratic formula is given as:
x = (-b ± √(b^2 - 4ac)) / (2a)
Comparing the equation 11x^2 - 742x + 6225 = 0 to ax^2 + bx + c = 0, we can determine the values of a, b, and c.
a = 11, b = -742, c = 6225
Step 6: Substituting the values of a, b, and c into the quadratic formula, we get:
x = (-(-742) ± √((-742)^2 - 4 * 11 * 6225)) / (2 * 11)
Simplifying further, we have:
x = (742 ± √(549124 - 273900)) / 22
x = (742 ± √275224) / 22
Step 7: Simplify the square root (√275224) as an irrational number or a simplified improper fraction if necessary.
x = (742 ± √(2 * 2 * 137 * 503)) / 22
x = (742 ± 2√(137 * 503)) / 22
The square root of the number inside the radical (√(137 * 503)) is already simplified, so we can leave it in that form.
Step 8: Simplify the expression by dividing the numerator and denominator by the greatest common divisor (GCD), which is 2 in this case.
x = (742 ± 2√(137 * 503)) / (22 * 2)
x = (371 ± √(137 * 503)) / 44
Therefore, the solution to the equation (√79 - 5x) = 6x + 4, while keeping the answer in simplified improper fraction form, is:
x = (371 ± √(137 * 503)) / 44