Solveeit Logo
Login

Question

Question: For what values of k, the equation \(9{x^2} - 24x + k = 0\) has equal roots? Find the roots....

For what values of k, the equation 9x2−24x+k=09{x^2} - 24x + k = 0 has equal roots? Find the roots.

Explanation

Solution

In the given question, we are required to solve for the value of k such that the equation 9x2−24x+k=09{x^2} - 24x + k = 0 has equal roots. We will first compare the given equation with the standard form of a quadratic equation ax2+bx+c=0a{x^2} + bx + c = 0 and then apply the quadratic formula to find the condition for equal roots of a quadratic equation.

Complete step-by-step solution:
In the given question, we are provided with the equation 9x2−24x+k=09{x^2} - 24x + k = 0.
Now, comparing the equation with standard form of a quadratic equation ax2+bx+c=0a{x^2} + bx + c = 0
Here,a=9a = 9, b=−24b = - 24 andc=kc = k.
Now, using the quadratic formula, we get the roots of the equation as:
x=(−b)±b2−4ac2ax = \dfrac{{( - b) \pm \sqrt {{b^2} - 4ac} }}{{2a}}
If both the roots of a quadratic equation are equal, then, we get,
x1=x2{x_1} = {x_2}
⇒(−b)+b2−4ac2a=(−b)−b2−4ac2a\Rightarrow \dfrac{{( - b) + \sqrt {{b^2} - 4ac} }}{{2a}} = \dfrac{{( - b) - \sqrt {{b^2} - 4ac} }}{{2a}}]
Cross multiplying the terms of equation and simplifying further, we get,
⇒b2−4ac=−b2−4ac\Rightarrow \sqrt {{b^2} - 4ac} = - \sqrt {{b^2} - 4ac}
Shifting all the terms to left side and dividing both sides of equation by two, we get,
⇒b2−4ac=0\Rightarrow \sqrt {{b^2} - 4ac} = 0
Now, we can substitute the values of a, b and c in the expression. So, we get,
⇒(−24)2−4(9)(k)=0\Rightarrow \sqrt {{{\left( { - 24} \right)}^2} - 4\left( 9 \right)\left( k \right)} = 0
⇒576−36k=0\Rightarrow \sqrt {576 - 36k} = 0
Factoring out 3636 from the expression and taking it out of the square root, we get,
⇒616−k=0\Rightarrow 6\sqrt {16 - k} = 0
Now, dividing both the sides of equation by six and squaring both sides of the equation, we get,
⇒16−k=0\Rightarrow \sqrt {16 - k} = 0
Squaring both sides of equation, we get,
⇒16−k=0\Rightarrow 16 - k = 0
Now, shifting the terms in equation using method of transposition, we get,
⇒k=16\Rightarrow k = 16
Hence, the value of k is 1616.

Note: We must know algebraic factorization and simplification rules in order to simplify the equation. One should know the expression for discriminant as b2−4ac{b^2} - 4ac for a quadratic equation
ax2+bx+c=0a{x^2} + bx + c = 0. Care should be taken while doing the calculations in order to get to the final answer.