Solve the system of equations. y = -27x - 23 y = x^2 - 13x + 26 Write the coordinates in exact form. Simplify all fractions and radicals. (______,______)

Question

Solve the system of equations. y = -27x - 23 y = x^2 - 13x + 26    Write the coordinates in exact form. Simplify all fractions and radicals. (______,______)

Bytelearn · Accepted Answer

Set Equations Equal: Since both equations are equal to y, we can set them equal to each other to find x. This gives us the equation -27x - 23 = x^2 - 13x + 26.Rearrange and Solve for x: Rearrange the equation to set it to zero and solve for x. This means we will move all terms to one side to get x^2 - 13x + 26 = 27x + 23. Simplifying this, we get x^2 - 40x + 3 = 0.Quadratic Formula: This is a quadratic equation, and we can solve for x by factoring, completing the square, or using the quadratic formula. The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a). For our equation, a = 1, b = -40, and c = 3.Calculate Discriminant: Calculate the discriminant Δ = b^2 - 4ac which is Δ = (-40)^2 - 4(1)(3). This gives us Δ = 1600 - 12, so Δ = 1588.Find Real Solutions: Since the discriminant is positive, we have two real solutions. Now we use the quadratic formula to find the values of x: x = (40 ± √1588) / 2. Simplifying under the radical, √1588 = √(4*397) = 2√397. So the solutions for x are x = (40 ± 2√397) / 2.Substitute x into Original Equation: Simplify the solutions for x by dividing by 2: x = 20 ± √397.Find y for x = 20 + √397: Now we substitute the values of x back into one of the original equations to find y. Let's use y = -27x - 23. First, we'll find y for x = 20 + √397. This gives us y = -27(20 + √397) - 23.Find y for x = 20 - √397: Simplify the expression for y: y = -540 - 27√397 - 23. Combining like terms, we get y = -563 - 27√397.Final Pairs of Solutions: Now we find y for x = 20 - √397. This gives us y = -27(20 - √397) - 23.Simplify the expression for y: y = -540 + 27√397 - 23. Combining like terms, we get y = -563 + 27√397.We now have two pairs of solutions for the system of equations: (20 + √397, -563 - 27√397) and (20 - √397, -563 + 27√397).

Solve the system of equations. $\newline$ $y = -27x - 23$ $\newline$ $y = x^2 - 13x + 26$ $\newline$ Write the coordinates in exact form. Simplify all fractions and radicals. $\newline$ (,)

Full solution

More problems from Solve a nonlinear system of equations

Related topics

Solve the system of equations.\newliney=−27x−23y = -27x - 23y=−27x−23\newliney=x2−13x+26y = x^2 - 13x + 26y=x2−13x+26\newlineWrite the coordinates in exact form. Simplify all fractions and radicals.\newline(______,______)

Solve the system of equations. $\newline$ $y = -27x - 23$ $\newline$ $y = x^2 - 13x + 26$ $\newline$ Write the coordinates in exact form. Simplify all fractions and radicals. $\newline$ (,)