Let's solve each quadratic equation step-by-step.
( x^2 - 4x + 3 = 0 )
Factoring: ( (x-1)(x-3) = 0 )
Solutions: ( x = 1, 3 )
( 2x^2 + 5x - 3 = 0 )
Using the quadratic formula:
( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} )
Here, ( a = 2, b = 5, c = -3 )
( x = \frac{{-5 \pm \sqrt{{5^2 - 4 \cdot 2 \cdot (-3)}}}}{2 \cdot 2} )
( x = \frac{{-5 \pm \sqrt{{49}}}}{4} )
( x = \frac{{-5 \pm 7}}{4} )
Solutions: ( x = \frac{1}{2}, -3 )
( 3x^2 - 12 = 0 )
Simplifying:
( 3x^2 = 12 )
( x^2 = 4 )
Solutions: ( x = 2, -2 )
( x^2 + 6x + 9 = 0 )
Factoring: ( (x+3)^2 = 0 )
Solution: ( x = -3 ) (double root)
( x^2 - 2x - 8 = 0 )
Factoring: ( (x-4)(x+2) = 0 )
Solutions: ( x = 4, -2 )
( 4x^2 + x - 10 = 0 )
Using the quadratic formula:
Here, ( a = 4, b = 1, c = -10 )
( x = \frac{{-1 \pm \sqrt{{1 + 160}}}}{8} )
( x = \frac{{-1 \pm 13}}{8} )
Solutions: ( x = \frac{3}{4}, -\frac{7}{4} )
( 5x^2 - 20x + 15 = 0 )
Dividing through by 5:
( x^2 - 4x + 3 = 0 ) (same as problem 1)
Solutions: ( x = 1, 3 )
( x^2 + 4x = 0 )
Factoring: ( x(x + 4) = 0 )
Solutions: ( x = 0, -4 )
( 9x^2 - 24x + 16 = 0 )
Using the quadratic formula:
Here, ( a = 9, b = -24, c = 16 )
( x = \frac{{24 \pm \sqrt{{576 - 576}}}}{2 \cdot 9} )
Solution: ( x = \frac{24}{18} = \frac{4}{3} ) (double root)
( 2x^2 - 3x + 1 = 0 )
Using the quadratic formula:
( a = 2, b = -3, c = 1 )
( x = \frac{{3 \pm \sqrt{{-1}}}}{4} )
Solutions: ( x = \frac{3}{4} \pm \frac{i}{2} )
( x^2 - 6 = 0 )
Rearranging:
( x^2 = 6 )
Solutions: ( x = \sqrt{6}, -\sqrt{6} )
( x^2 + 2x + 1 = 0 )
Factoring: ( (x+1)^2 = 0 )
Solution: ( x = -1 ) (double root)
( 7x^2 + 2x - 14 = 0 )
Using the quadratic formula:
( a = 7, b = 2, c = -14 )
( x = \frac{{-2 \pm \sqrt{{4 + 392}}}}{14} )
( x = \frac{{-2 \pm 20}}{14} )
Solutions: ( x = \frac{9}{7}, -\frac{11}{7} )
( x^2 - 10x + 25 = 0 )
Factoring: ( (x-5)^2 = 0 )
Solution: ( x = 5 ) (double root)
( 6x^2 + 5 = 0 )
Rearranging:
( 6x^2 = -5 )
Solutions: ( x = \pm \sqrt{-\frac{5}{6}} = \pm \frac{i\sqrt{30}}{6} )
( x^2 + 5x - 14 = 0 )
Using the quadratic formula:
( a = 1, b = 5, c = -14 )
( x = \frac{{-5 \pm \sqrt{{25 + 56}}}}{2} )
( x = \frac{{-5 \pm 11}}{2} )
Solutions: ( x = 3, -8 )
( 8x - 2x + 1 = 0 )
Simplifying:
( 6x + 1 = 0 )
Solution: ( x = -\frac{1}{6} )
( 3x^2 + 12x = 0 )
Factoring: ( 3x(x + 4) = 0 )
Solutions: ( x = 0, -4 )
( x^2 - 3x - 4 = 0 )
Factoring: ( (x - 4)(x + 1) = 0 )
Solutions: ( x = 4, -1 )
( 10x^2 + 1 - 20 = 0 )
Rearranging:
( 10x^2 - 19 = 0 )
Solutions: ( x = \pm \sqrt{\frac{19}{10}} )
( x^2 + 8 = 0 )
Rearranging:
( x^2 = -8 )
Solutions: ( x = \pm 2i\sqrt{2} )
( 2x^2 - 8x + 6 = 0 )
Using the quadratic formula:
( a = 2, b = -8, c = 6 )
( x = \frac{{8 \pm \sqrt{{64 - 48}}}}{4} )
Solutions: ( x = 3, 1.5 )
( 4x^2 + 9x + 2 = 0 )
Using the quadratic formula:
( a = 4, b = 9, c = 2 )
( x = \frac{{-9 \pm \sqrt{{81 - 32}}}}{8} )
Solutions: ( x = \frac{-9 \pm \sqrt{49}}{8} = \frac{-9 \pm 7}{8} )
Solutions: ( x = -\frac{1}{4}, -2 )
( 5x^2 - 5x = 0 )
Factoring: ( 5x(x - 1) = 0 )
Solutions: ( x = 0, 1 )
( x^2 + x - 6 = 0 )
Factoring: ( (x - 2)(x + 3) = 0 )
Solutions: ( x = 2, -3 )
( 3x - 27 = 0 )
Rearranging:
( 3x = 27 )
Solution: ( x = 9 )
( 2x^2 + 3x + 1 = 0 )
Using the quadratic formula:
( a = 2, b = 3, c = 1 )
( x = \frac{{-3 \pm \sqrt{9 - 8}}}{4} )
Solutions: ( x = -1, -\frac{1}{2} )
( x^2 - 8x + 16 = 0 )
Factoring: ( (x - 4)^2 = 0 )
Solution: ( x = 4 ) (double root)
( x^2 + 10x + 25 = 0 )
Factoring: ( (x + 5)^2 = 0 )
Solution: ( x = -5 ) (double root)
( 6x^2 - 12x = 0 )
Factoring: ( 6x(x - 2) = 0 )
Solutions: ( x = 0, 2 )
This completes the solutions for the equations provided.
