The sum of the roots of the equation four 𝑥 squared plus 𝑘𝑥 minus four equals zero is negative one. Find the value of 𝑘 and the solution set of the equation.
We begin by noticing that our quadratic equation is written in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero, where 𝑎 is equal to four, 𝑏 is equal to 𝑘, and 𝑐 is equal to negative four. We know that the sum of the roots of any equation of this type is equal to negative 𝑏 over 𝑎 and the product of the roots is equal to 𝑐 over 𝑎. In this question, we are told that the sum of the roots is equal to negative one. This means that negative 𝑘 over four is equal to negative one. Multiplying both sides of this equation by four, we have negative 𝑘 is equal to negative four. We can then multiply or divide both sides of this equation by negative one to give us 𝑘 is equal to four. This means that our quadratic equation can be rewritten as four 𝑥 squared plus four 𝑥 minus four equals zero.
We can solve this using the quadratic formula. However, it is worth noticing that all three terms on the left-hand side are divisible by four. This means that we can rewrite the equation as 𝑥 squared plus 𝑥 minus one equals zero. The quadratic formula states that 𝑥 is equal to negative 𝑏 plus or minus the square root of 𝑏 squared minus four 𝑎𝑐 all divided by two 𝑎. And in the equation 𝑥 squared plus 𝑥 minus one equals zero, our values of 𝑎, 𝑏, and 𝑐 are one, one, and negative one. Substituting these into the formula, we have 𝑥 is equal to negative one plus or minus the square root of one squared minus four multiplied by one multiplied by negative one all divided by two multiplied by one. This simplifies to negative one plus or minus the square root of five all divided by two.
The solution set to our quadratic equation contains the two values negative one plus root five divided by two and negative one minus root five divided by two.