### Video Transcript

By using the quadratic formula, solve the equation two 𝑥 squared plus three 𝑥 is equal to seven. Give your answers correct to two decimal places.

We recall that we can solve any quadratic equation of the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero, where 𝑎, 𝑏, and 𝑐 are constants and 𝑎 is not equal to zero, by using the quadratic formula. This 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. We obtain two solutions, one using the positive sign and one from the negative sign.

In this question, we begin by subtracting seven from both sides of our equation. This gives us two 𝑥 squared plus three 𝑥 minus seven equals zero. Our equation is now written in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero, where the values of 𝑎, 𝑏, and 𝑐 are two, three, and negative seven, respectively.

Substituting these values into the quadratic formula gives us 𝑥 is equal to negative three plus or minus the square root of three squared minus four multiplied by two multiplied by negative seven all divided by two multiplied by two. Three squared is equal to nine. Multiplying four by two and then by negative seven gives us negative 56. On the numerator, two multiplied by two is equal to four. 𝑥 is therefore equal to negative three plus or minus the square root of nine minus negative 56 all divided by four. This once again simplifies to 𝑥 is equal to negative three plus or minus the square root of 65 all divided by four. This gives us two possible solutions. Either 𝑥 is equal to negative three plus the square root of 65 all divided by four or 𝑥 is equal to negative three minus the square root of 65 all divided by four.

Typing the first calculation into our calculator gives us 1.2655 and so on, and rounding this to two decimal places gives us 1.27. The second calculation rounded to two decimal places is negative 2.77. Using the quadratic formula, the solutions to the equation two 𝑥 squared plus three 𝑥 equals seven to two decimal places are 𝑥 is equal to 1.27 and 𝑥 is equal to negative 2.77.