Karim is solving the quadratic equation x^2+5x−24=0 by grouping. His work is shown below. Determine the first step at which Karim makes an error. Step 1: x^2+8x+(−3x)−24=0 Step 2: (x^2+8x)+(−3x−24)=0 Step 3: x(x+8)+3(x+8)=0 Step 4: (x+8)(x+3)=0 Step 5: x+8=0 or x+3=0 x=−8 or x=−3
Karim makes an error in Step 1. The correct grouping should be x^2+3x+8x−24=0.
The first step at which Karim makes an error is Step 2: (x^2+8x)+(−3x−24)=0. In this step, Karim incorrectly groups the terms by splitting the constant term -24 as -3x and -24 instead of -8x and -24. The correct grouping would be: (x^2+8x)+(−3x−24)=0.
Karim made an error in Step 3. Let's understand how to solve the quadratic equation correctly using grouping.
To solve the quadratic equation x^2 + 5x - 24 = 0 by grouping, we need to factorize the equation. Here's the correct step-by-step process:
Step 1: Write down the quadratic equation: x^2 + 5x - 24 = 0
Step 2: Break down the middle term (5x) into two terms whose coefficients multiply to give the product of the quadratic term coefficient (1) and the constant term (-24). In this case, we need to find two numbers whose product is -24 and sum is 5. The numbers are 8 and -3.
Step 3: Group the terms accordingly: x^2 + 8x - 3x - 24 = 0
Step 4: Factor by grouping. Now, factor the expression by grouping the terms as follows:
(x^2 + 8x) - (3x + 24) = 0
Step 5: Factor out the common factor from each group:
x(x + 8) - 3(x + 8) = 0
Step 6: Notice that (x + 8) is common in both terms. Factor it out:
(x + 8)(x - 3) = 0
Step 7: Set each factor equal to zero and solve for x:
x + 8 = 0 or x - 3 = 0
Step 8: Solve for x in each equation:
x = -8 or x = 3
Therefore, Karim made an error in Step 3 by incorrectly distributing the 3 to both terms within parentheses. The corrected expression should be x(x + 8) - 3(x + 8) = 0.