One of the steps needed to solve a quadratic equation by grouping is to split the middle term as a sum of two terms. Given the equation x^2−13x+42=0 , which two numbers would be needed to complete this step? Enter the lesser value first.(1 point) x^2−13x+42=0 x^2+ x+ x+42=0
To solve the equation by grouping, we need to split the middle term -13x as a sum of two terms. The two terms will also have a product of 42.
The factors of 42 are: 1, 2, 3, 6, 7, 14, 21, 42
When splitting -13x, we need to find two numbers whose sum is -13 and whose product is 42.
The two numbers that satisfy these conditions are -6 and -7.
Therefore, we would split the middle term -13x as -6x and -7x.
The equation becomes:
x^2 - 6x - 7x + 42 = 0
To split the middle term x^2−13x+42=0, we need to find two numbers whose sum is -13 (the coefficient of the x term) and whose product is the product of the coefficient of the x^2 term (which is 1) and the constant term (which is 42).
The two numbers that satisfy these criteria are -6 and -7 (since -6 + -7 = -13 and -6 * -7 = 42).
Therefore, to split the middle term, we rewrite the equation as:
x^2 - 6x - 7x + 42 = 0
To solve a quadratic equation by grouping, we want to split the middle term as a sum of two terms.
For the equation x^2−13x+42=0, the middle term is -13x. To split -13x as a sum of two terms, we need to find two numbers that multiply to give 42 (the product of the coefficient of x^2 and the constant term) and add up to -13 (the coefficient of x).
To find these two numbers, we can use factoring or trial and error. Let's use trial and error in this case:
List all the factor pairs of 42: (1, 42), (2, 21), (3, 14), (6, 7)
From these factor pairs, we want the pair that adds up to -13. In this case, (-3, -14) is the pair that satisfies this condition.
So, we can split -13x as -3x - 14x:
x^2 - 3x - 14x + 42 = 0
Now, we can group the terms:
(x^2 - 3x) + (-14x + 42) = 0
Next, we can factor out common terms from each group:
x(x - 3) - 14(x - 3) = 0
Notice that we have a common factor of (x - 3), so we can factor it out:
(x - 3)(x - 14) = 0
Now, we can set each factor equal to zero and solve for x:
x - 3 = 0 --> x = 3
x - 14 = 0 --> x = 14
Therefore, the two numbers needed to complete the step of splitting the middle term are -3 and -14, with the lesser value being -14.