If one of the roots of quadratic equation X^2 + kx + 54 = 0 is -6 then complete the following activity to find the value of 'k'. 1) Solve the following quadratic equation by factor... If one of the roots of quadratic equation X^2 + kx + 54 = 0 is -6 then complete the following activity to find the value of 'k'. 1) Solve the following quadratic equation by factorization method. 3p^2 + 8p + 5 = 0 2) If one of the roots of quadratic equation X^2 - kx - 15 = 0 is -3 then find the value of 'k'. 3) If the Roots of a quadratic equation are 4 and -5 then form the quadratic equation. 4) If roots of a quadratic equation 3y^2 + ky + 12 = 0 are real and equal then find the value of 'k'. 5) Roots of a quadratic equation are 5 and -4 then form the quadratic equation.
Understand the Problem
The question presents several tasks related to quadratic equations, including solving, finding roots, and determining values for variables involved in those equations. It covers both solving quadratic equations and determining if particular values are roots of specified equations.
Answer
The value of \( k \) is \( 15 \).
Answer for screen readers
The value of ( k ) is ( 15 ).
Steps to Solve
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Write down the given quadratic equation
You need to work with the equation ( X^2 + kX + 54 = 0 ), where one of the roots is given as (-6).
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Substitute the known root into the equation
Replace ( X ) with (-6): $$ (-6)^2 + k(-6) + 54 = 0 $$
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Simplify the equation
Calculate ((-6)^2 = 36): $$ 36 - 6k + 54 = 0 $$
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Combine like terms
Combine ( 36 + 54 ): $$ 90 - 6k = 0 $$
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Solve for ( k )
Rearrange the equation: $$ 90 = 6k $$ Divide by 6: $$ k = 15 $$
The value of ( k ) is ( 15 ).
More Information
In this problem, we used the fact that if a number is a root of a quadratic equation, then substituting this number into the equation should yield zero. By using this property, we could find the coefficient ( k ).
Tips
- Forgetting to square the root correctly, which can lead to incorrect simplifications.
- Misplacing the negative sign when substituting the root into the equation. This could alter all steps following the substitution.
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