Find the solution set of the equation five 𝑥 multiplied by 𝑥 minus six minus three multiplied by 𝑥 plus four plus three equals zero, giving values to one decimal place.
In order to solve this equation, we will begin by distributing the parentheses or expanding the brackets. Five 𝑥 multiplied by 𝑥 is five 𝑥 squared, and five 𝑥 multiplied by negative six is negative 30𝑥. Negative three multiplied by 𝑥 is negative three 𝑥, and negative three multiplied by four is negative 12. Our equation simplifies to five 𝑥 squared minus 30𝑥 minus three 𝑥 minus 12 plus three equals zero. Our next step is to group or collect the like terms. This gives us five 𝑥 squared minus 33𝑥 minus nine equals zero.
We now have a quadratic equation written in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero. For any equation of this type, where 𝑎, 𝑏, and 𝑐 are constants and 𝑎 is nonzero, we can use the quadratic formula to solve it. 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. Our equation has values of 𝑎, 𝑏, and 𝑐 of five, negative 33, and negative nine, respectively. Substituting these values into the quadratic formula, we have 𝑥 is equal to negative negative 33 plus or minus the square root of negative 33 squared minus four multiplied by five multiplied by negative nine.
Negative negative 33 is the same as positive 33. And when typing negative 33 squared in our calculator, it is important that we put the negative 33 in brackets. This is important as if we just typed negative 33 squared, we would get in answer of negative 1089. However, we know that when we square a negative number, we get a positive answer. As a result, the correct way to type this into the calculator is with the negative 33 in brackets or parentheses. The expression under the square root therefore simplifies to 1269. We are left with 𝑥 is equal to 33 plus or minus the square root of 1269 all divided by 10. This in turn gives us two possible values of 𝑥. When we add the square root of 1269, our value of 𝑥 is 6.862 and so on. And when we subtract the square root of 1269, we get 𝑥 is equal to negative 0.262 and so on.
We are asked to give our solutions to one decimal place. This means that the solution set of the equation five 𝑥 multiplied by 𝑥 minus six minus three multiplied by 𝑥 plus four plus three equals zero contains the values 6.9 and negative 0.3 correct to one decimal place.