Factor Quadratic Equation: Solve the given quadratic equation for x. The given equation is 3x2−18x−15=0. To solve for x, we can either factor the quadratic, complete the square, or use the quadratic formula. Let's try to factor it first.
Find Roots: Factor the quadratic equation.We look for two numbers that multiply to (3×−15)=−45 and add up to −18. These numbers are −15 and −3.So we can write the equation as 3x2−15x−3x−15=0.Grouping the terms, we get (3x2−15x)−(3x+15)=0.Factoring out the common factors, we get 3x(x−5)−3(x+5)=0.Taking out 3 as a common factor, we get 3(x−5)(x+5)=0.
Substitute Roots: Find the roots of the equation.Setting each factor equal to zero gives us the possible solutions for x:x−5=0 or x+5=0.Therefore, x=5 or x=−5.
Verify Solutions: Substitute the roots into the expression x2−6x.We need to find the value of x2−6x for each root and see if they are equal since the question does not specify which root to use.For x=5:(5)2−6(5)=25−30=−5.For x=−5:(−5)2−6(−5)=25+30=55.The values are not the same, which means we need to check the original equation to see if both roots are valid.
Verify Solutions: Substitute the roots into the expression x2−6x. We need to find the value of x2−6x for each root and see if they are equal since the question does not specify which root to use. For x=5: (5)2−6(5)=25−30=−5. For x=−5: (−5)2−6(−5)=25+30=55. The values are not the same, which means we need to check the original equation to see if both roots are valid.Verify the roots in the original equation. For x=5: 3(5)2−18(5)−15=3(25)−90−15=75−90−15=−30−15=−45, which is not equal to 0. This means x=5 is not a valid solution. For x=−5: x2−6x1, which is not equal to 0. This means x=−5 is not a valid solution either. There seems to be a mistake in our calculations or in the factoring process. We need to re-evaluate our steps to find the error.
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