b) (2x-5)^2+(x-2)(x+2)=x+(x+7)^2 c) (x-1)^2+x(x+1)=2x-(x+3)^2

b) $(2x-5)^{2}+(x-2)(x+2)=x+(x+7)^{2}$<br /><br />To solve this equation, we need to expand and simplify both sides of the equation.<br /><br />Expanding the left side:<br />$(2x-5)^{2}+(x-2)(x+2)$<br />$= (4x^2 - 20x + 25) + (x^2 - 4)$<br />$= 5x^2 - 20x + 21$<br /><br />Expanding the right side:<br />$x+(x+7)^{2}$<br />$= x + (x^2 + 14x + 49)$<br />$= x^2 + 15x + 49$<br /><br />Setting the two expressions equal to each other:<br />$5x^2 - 20x + 21 = x^2 + 15x + 49$<br /><br />Simplifying the equation:<br />$4x^2 - 35x - 28 = 0$<br /><br />Solving the quadratic equation:<br />$x = \frac{35 \pm \sqrt{35^2 - 4(4)(-28)}}{2(4)}$<br />$x = \frac{35 \pm \sqrt{1225 + 448}}{8}$<br />$x = \frac{35 \pm \sqrt{1673}}{8}$<br />$x = \frac{35 \pm 40.8}{8}$<br /><br />Therefore, the solutions are:<br />$x = \frac{35 + 40.8}{8} = 9.85$<br />$x = \frac{35 - 40.8}{8} = -0.85$<br /><br />c) $(x-1)^{2}+x(x+1)=2x-(x+3)^{2}$<br /><br />Expanding the left side:<br />$(x-1)^{2}+x(x+1)$<br />$= (x^2 - 2x + 1) + (x^2 + x)$<br />$= 2x^2 - x + 1$<br /><br />Expanding the right side:<br />$2x-(x+3)^{2}$<br />$= 2x - (x^2 + 6x + 9)$<br />$= 2x - x^2 - 6x - 9$<br />$= -x^2 - 4x - 9$<br /><br />Setting the two expressions equal to each other:<br />$2x^2 - x + 1 = -x^2 - 4x - 9$<br /><br />Simplifying the equation:<br />$3x^2 + 3x + 10 = 0$<br /><br />Solving the quadratic equation:<br />$x = \frac{-3 \pm \sqrt{3^2 - 4(3)(10)}}{2(3)}$<br />$x = \frac{-3 \pm \sqrt{9 - 120}}{6}$<br />$x = \frac{-3 \pm \sqrt{-111}}{6}$<br /><br />Since the discriminant is negative, there are no real solutions for this equation.