Solving Quadratic Equations Using Patterns Practice

Question

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Question
Supply the number so that the quadratic equation below has 6 and 8 as its solutions.(1 point)
x2−14x+
=0
Remaining Attempts : 3

Answer 1

To find the missing number in the quadratic equation \( x^2 - 14x + __ = 0 \) such that the solutions are 6 and 8, we can use the fact that a quadratic can be expressed in factored form as:

\[ (x - r_1)(x - r_2) = 0 \]

where \( r_1 \) and \( r_2 \) are the solutions (in this case, 6 and 8).

Using this formula, we can express the quadratic equation:

\[ (x - 6)(x - 8) = 0 \]

Now, we will expand this product:

\[ x^2 - 8x - 6x + 48 = x^2 - 14x + 48 \]

This shows that the constant term in the equation must be 48. Thus, filling in the blank, we get:

\[ x^2 - 14x + 48 = 0 \]

The missing number is 48.