WEBVTT
00:00:00.290 --> 00:00:11.070
The sum of the roots of the equation four 𝑥 squared plus 𝑘𝑥 minus four equals zero is negative one.
00:00:11.930 --> 00:00:16.500
Find the value of 𝑘 and the solution set of the equation.
00:00:17.830 --> 00:00:35.020
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.
00:00:35.860 --> 00:00:46.130
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.
00:00:47.160 --> 00:00:52.280
In this question, we are told that the sum of the roots is equal to negative one.
00:00:53.080 --> 00:00:58.400
This means that negative 𝑘 over four is equal to negative one.
00:00:59.310 --> 00:01:05.880
Multiplying both sides of this equation by four, we have negative 𝑘 is equal to negative four.
00:01:06.540 --> 00:01:14.870
We can then multiply or divide both sides of this equation by negative one to give us 𝑘 is equal to four.
00:01:15.940 --> 00:01:25.340
This means that our quadratic equation can be rewritten as four 𝑥 squared plus four 𝑥 minus four equals zero.
00:01:26.750 --> 00:01:29.670
We can solve this using the quadratic formula.
00:01:30.130 --> 00:01:36.530
However, it is worth noticing that all three terms on the left-hand side are divisible by four.
00:01:37.310 --> 00:01:44.960
This means that we can rewrite the equation as 𝑥 squared plus 𝑥 minus one equals zero.
00:01:45.820 --> 00:01:57.980
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.
00:01:58.970 --> 00:02:10.870
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.
00:02:11.810 --> 00:02:29.590
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.
00:02:30.610 --> 00:02:38.590
This simplifies to negative one plus or minus the square root of five all divided by two.
00:02:39.780 --> 00:02:53.700
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.