Example Given \( m=x(x-y)^{2} \). Determine the value of \( m \) if \( x y^{2}=4 \) and \( x^{2}-2 x^{2} y=3 \)

Given the equation \( m=x(x-y)^{2} \), we need to find the value of \( m \) when \( x y^{2}=4 \) and \( x^{2}-2 x^{2} y=3 \). First, let's simplify the given equations: 1. \( x y^{2}=4 \) 2. \( x^{2}-2 x^{2} y=3 \) From equation 1, we can express \( y \) in terms of \( x \): \( y = \frac{4}{x} \) Substitute \( y = \frac{4}{x} \) into equation 2: \( x^{2} - 2x^{2} \left(\frac{4}{x}\right) = 3 \) \( x^{2} - 8x = 3 \) \( x^{2} - 8x - 3 = 0 \) Now, we have a quadratic equation in terms of \( x \). We can solve this equation to find the values of \( x \) and then substitute these values back into the equation for \( m \) to find the value of \( m \). Solve the quadratic equation by following steps: - step0: Solve using the quadratic formula: \(x^{2}-8x-3=0\) - step1: Solve using the quadratic formula: \(x=\frac{8\pm \sqrt{\left(-8\right)^{2}-4\left(-3\right)}}{2}\) - step2: Simplify the expression: \(x=\frac{8\pm \sqrt{76}}{2}\) - step3: Simplify the expression: \(x=\frac{8\pm 2\sqrt{19}}{2}\) - step4: Separate into possible cases: \(\begin{align}&x=\frac{8+2\sqrt{19}}{2}\\&x=\frac{8-2\sqrt{19}}{2}\end{align}\) - step5: Simplify the expression: \(\begin{align}&x=4+\sqrt{19}\\&x=\frac{8-2\sqrt{19}}{2}\end{align}\) - step6: Simplify the expression: \(\begin{align}&x=4+\sqrt{19}\\&x=4-\sqrt{19}\end{align}\) - step7: Rewrite: \(x_{1}=4-\sqrt{19},x_{2}=4+\sqrt{19}\) The solutions to the quadratic equation \( x^{2} - 8x - 3 = 0 \) are \( x_{1}=4-\sqrt{19} \) and \( x_{2}=4+\sqrt{19} \). Now, we need to substitute these values of \( x \) back into the equation for \( m \) to find the value of \( m \). Substitute \( x = 4-\sqrt{19} \) into the equation for \( m \): \( m = (4-\sqrt{19})(4-\sqrt{19}-\frac{4}{4-\sqrt{19}})^{2} \) Substitute \( x = 4+\sqrt{19} \) into the equation for \( m \): \( m = (4+\sqrt{19})(4+\sqrt{19}-\frac{4}{4+\sqrt{19}})^{2} \) Let's calculate the values of \( m \) for both \( x \) values. Calculate the value by following steps: - step0: Calculate: \(\left(4-\sqrt{19}\right)\left(4-\sqrt{19}-\frac{4}{4-\sqrt{19}}\right)^{2}\) - step1: Subtract the numbers: \(\left(4-\sqrt{19}\right)\left(\frac{31-8\sqrt{19}}{4-\sqrt{19}}\right)^{2}\) - step2: Apply the distributive property: \(4\left(\frac{31-8\sqrt{19}}{4-\sqrt{19}}\right)^{2}-\sqrt{19}\times \left(\frac{31-8\sqrt{19}}{4-\sqrt{19}}\right)^{2}\) - step3: Multiply the terms: \(\frac{3212+224\sqrt{19}}{9}-\sqrt{19}\times \left(\frac{31-8\sqrt{19}}{4-\sqrt{19}}\right)^{2}\) - step4: Multiply: \(\frac{3212+224\sqrt{19}}{9}-\left(\frac{31-8\sqrt{19}}{4-\sqrt{19}}\right)^{2}\sqrt{19}\) - step5: Evaluate the power: \(\frac{3212+224\sqrt{19}}{9}-\frac{803\sqrt{19}+1064}{9}\) - step6: Transform the expression: \(\frac{3212+224\sqrt{19}-\left(803\sqrt{19}+1064\right)}{9}\) - step7: Calculate: \(\frac{2148-579\sqrt{19}}{9}\) - step8: Rewrite the expression: \(\frac{3\left(716-193\sqrt{19}\right)}{9}\) - step9: Reduce the fraction: \(\frac{716-193\sqrt{19}}{3}\) Calculate or simplify the expression \( (4+\sqrt{19})(4+\sqrt{19}-\frac{4}{4+\sqrt{19}})^{2} \). Calculate the value by following steps: - step0: Calculate: \(\left(4+\sqrt{19}\right)\left(4+\sqrt{19}-\frac{4}{4+\sqrt{19}}\right)^{2}\) - step1: Calculate: \(\left(4+\sqrt{19}\right)\left(\frac{31+8\sqrt{19}}{4+\sqrt{19}}\right)^{2}\) - step2: Apply the distributive property: \(4\left(\frac{31+8\sqrt{19}}{4+\sqrt{19}}\right)^{2}+\sqrt{19}\times \left(\frac{31+8\sqrt{19}}{4+\sqrt{19}}\right)^{2}\) - step3: Multiply the terms: \(\frac{3212-224\sqrt{19}}{9}+\sqrt{19}\times \left(\frac{31+8\sqrt{19}}{4+\sqrt{19}}\right)^{2}\) - step4: Multiply the numbers: \(\frac{3212-224\sqrt{19}}{9}+\frac{\left(31+8\sqrt{19}\right)^{2}}{\left(4+\sqrt{19}\right)^{2}}\times \sqrt{19}\) - step5: Evaluate the power: \(\frac{3212-224\sqrt{19}}{9}+\frac{2177\sqrt{19}+9424}{35+8\sqrt{19}}\) - step6: Reduce fractions to a common denominator: \(\frac{\left(3212-224\sqrt{19}\right)\left(35+8\sqrt{19}\right)}{9\left(35+8\sqrt{19}\right)}+\frac{\left(2177\sqrt{19}+9424\right)\times 9}{\left(35+8\sqrt{19}\right)\times 9}\) - step7: Multiply the terms: \(\frac{\left(3212-224\sqrt{19}\right)\left(35+8\sqrt{19}\right)}{315+72\sqrt{19}}+\frac{\left(2177\sqrt{19}+9424\right)\times 9}{\left(35+8\sqrt{19}\right)\times 9}\) - step8: Multiply the terms: \(\frac{\left(3212-224\sqrt{19}\right)\left(35+8\sqrt{19}\right)}{315+72\sqrt{19}}+\frac{\left(2177\sqrt{19}+9424\right)\times 9}{315+72\sqrt{19}}\) - step9: Transform the expression: \(\frac{\left(3212-224\sqrt{19}\right)\left(35+8\sqrt{19}\right)+\left(2177\sqrt{19}+9424\right)\times 9}{315+72\sqrt{19}}\) - step10: Multiply the terms: \(\frac{78372+17856\sqrt{19}+\left(2177\sqrt{19}+9424\right)\times 9}{315+72\sqrt{19}}\) - step11: Multiply the terms: \(\frac{78372+17856\sqrt{19}+19593\sqrt{19}+84816}{315+72\sqrt{19}}\) - step12: Add the numbers: \(\frac{163188+37449\sqrt{19}}{315+72\sqrt{19}}\) - step13: Rewrite the expression: \(\frac{9\left(18132+4161\sqrt{19}\right)}{9\left(35+8\sqrt{19}\right)}\) - step14: Reduce the fraction: \(\frac{18132+4161\sqrt{19}}{35+8\sqrt{19}}\) - step15: Multiply by the Conjugate: \(\frac{\left(18132+4161\sqrt{19}\right)\left(35-8\sqrt{19}\right)}{\left(35+8\sqrt{19}\right)\left(35-8\sqrt{19}\right)}\) - step16: Multiply the numbers: \(\frac{\left(18132+4161\sqrt{19}\right)\left(35-8\sqrt{19}\right)}{9}\) - step17: Factor the expression: \(\frac{\left(6044+1387\sqrt{19}\right)\left(35-8\sqrt{19}\right)\times 3}{3\times 3}\) - step18: Reduce the fraction: \(\frac{\left(6044+1387\sqrt{19}\right)\left(35-8\sqrt{19}\right)}{3}\) - step19: Multiply the numbers: \(\frac{716+193\sqrt{19}}{3}\) The value of \( m \) when \( x = 4-\sqrt{19} \) is approximately -41.755832. The value of \( m \) when \( x = 4+\sqrt{19} \) is approximately 519.089165. Therefore, the value of \( m \) is approximately -41.755832 when \( x = 4-\sqrt{19} \) and approximately 519.089165 when \( x = 4+\sqrt{19} \).