Question

Find \(k\) so that the equation \(k x^{2}+x+k=0\) has a repeated real solution.

Short answer

The values of \(k\) are \(\frac{1}{2}\) or \(-\frac{1}{2}\).

Step-by-step solution

Identify the condition for repeated real solutions

For a quadratic equation to have a repeated real solution, the discriminant must be equal to zero. The general form of a quadratic equation is \(ax^2 + bx + c = 0\), and the discriminant is given by \(\triangle = b^2 - 4ac\). It must hold that \(\triangle = 0\) for repeated real solutions.

Determine the coefficients

Compare the given equation \(k x^{2}+x+k=0\) with the standard form \(ax^2 + bx + c = 0\). Here, \(a = k\), \(b = 1\), and \(c = k\).

Calculate the discriminant

Substitute \(a\), \(b\), and \(c\) into the discriminant formula \(\triangle = b^2 - 4ac\). This gives \(\triangle = 1^2 - 4(k)(k)\).Simplify to get \(\triangle = 1 - 4k^2\).

Set the discriminant to zero

For repeated real solutions, set the discriminant equal to zero: \(1 - 4k^2 = 0\).

Solve for \(k\)

Solve the equation \(1 - 4k^2 = 0\) for \(k\).Adding \(4k^2\) to both sides results in \(1 = 4k^2\).Divide both sides by \(4\) to get \(k^2 = \frac{1}{4}\).Take the square root of both sides to find \(k = \frac{1}{2}\) or \(k = -\frac{1}{2}\).

Key concepts

discriminant

The discriminant is a crucial part of solving quadratic equations. It is found using the formula \[ \triangle = b^2 - 4ac \]. This small but powerful expression tells us about the nature of the roots of the quadratic equation. For example, if the discriminant is positive, we get two distinct real solutions. If it’s zero, we have a repeated real solution, which means both roots are the same. And if the discriminant is negative, we get complex solutions. So, remember, checking the discriminant can tell a lot about the solutions without actually solving the quadratic equation. In this exercise, the discriminant helps us find the right value of \(k\) so that our quadratic equation has a repeated real solution.

repeated real solution

A repeated real solution, also known as a double root, happens when a quadratic equation yields the same solution twice. This occurs specifically when the discriminant \( \triangle \) is zero. To explore this, let's look at the problem we have: \(k x^2 + x + k = 0\). We need to ensure the discriminant is zero to get this double root. For our quadratic form, \(a = k\), \(b = 1\), and \(c = k\). Using these coefficients in the discriminant formula, we get \( \triangle = 1^2 - 4k^2 \). Setting it to zero, we solve \(1 - 4k^2 = 0 \), indicating a repeated real solution.

solving for coefficients

When solving quadratic equations, identifying and working with the coefficients is key. For the given problem \(k x^2 + x + k = 0\), we start by comparing it with the standard form of a quadratic equation \(a x^2 + b x + c = 0\). Here, \(a\) is the coefficient of \( x^2 \), \(b\) is the coefficient of \( x \), and \(c\) is the constant term. In our problem, we see that \(a = k\), \(b = 1\), and \(c = k\). Plugging these into the discriminant formula \(b^2 - 4ac\), and simplifying, we exactly match the condition for repeated real solutions. This clear understanding of coefficients helps to streamline the whole problem-solving process.