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If 3x^(2)-18 x-15=0, what is the value of x^(2)-6x ?
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If 3x218x15=0 3 x^{2}-18 x-15=0 , what is the value of x26x x^{2}-6 x ?\newline \square

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Q. If 3x218x15=0 3 x^{2}-18 x-15=0 , what is the value of x26x x^{2}-6 x ?\newline \square
  1. Factor Quadratic Equation: Solve the given quadratic equation for xx. The given equation is 3x218x15=03x^2 - 18x - 15 = 0. To solve for xx, we can either factor the quadratic, complete the square, or use the quadratic formula. Let's try to factor it first.
  2. Find Roots: Factor the quadratic equation.\newlineWe look for two numbers that multiply to (3×15)=45(3 \times -15) = -45 and add up to 18-18. These numbers are 15-15 and 3-3.\newlineSo we can write the equation as 3x215x3x15=03x^2 - 15x - 3x - 15 = 0.\newlineGrouping the terms, we get (3x215x)(3x+15)=0(3x^2 - 15x) - (3x + 15) = 0.\newlineFactoring out the common factors, we get 3x(x5)3(x+5)=03x(x - 5) - 3(x + 5) = 0.\newlineTaking out 33 as a common factor, we get 3(x5)(x+5)=03(x - 5)(x + 5) = 0.
  3. Substitute Roots: Find the roots of the equation.\newlineSetting each factor equal to zero gives us the possible solutions for xx:\newlinex5=0x - 5 = 0 or x+5=0x + 5 = 0.\newlineTherefore, x=5x = 5 or x=5x = -5.
  4. Verify Solutions: Substitute the roots into the expression x26xx^2 - 6x.\newlineWe need to find the value of x26xx^2 - 6x for each root and see if they are equal since the question does not specify which root to use.\newlineFor x=5x = 5:\newline(5)26(5)=2530=5(5)^2 - 6(5) = 25 - 30 = -5.\newlineFor x=5x = -5:\newline(5)26(5)=25+30=55(-5)^2 - 6(-5) = 25 + 30 = 55.\newlineThe values are not the same, which means we need to check the original equation to see if both roots are valid.
  5. Verify Solutions: Substitute the roots into the expression x26xx^2 - 6x. We need to find the value of x26xx^2 - 6x for each root and see if they are equal since the question does not specify which root to use. For x=5x = 5: (5)26(5)=2530=5(5)^2 - 6(5) = 25 - 30 = -5. For x=5x = -5: (5)26(5)=25+30=55(-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=5x = 5: 3(5)218(5)15=3(25)9015=759015=3015=453(5)^2 - 18(5) - 15 = 3(25) - 90 - 15 = 75 - 90 - 15 = -30 - 15 = -45, which is not equal to 00. This means x=5x = 5 is not a valid solution. For x=5x = -5: x26xx^2 - 6x11, which is not equal to 00. This means x=5x = -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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