Check whether the following are quadratic equations:
(i) \((x + 1)^2 = 2(x - 3)\)
(ii) \(x^2 - 2x = (-2)(3 - x)\)
(iii) \((x - 2)(x + 1) = (x - 1)(x + 3)\)
(iv) \((x - 3)(2x + 1) = x(x + 5)\)
(v) \((2x - 1)(x - 3) = (x + 5)(x - 1)\)
(vi) \(x^2 + 3x + 1 = (x - 2)^2\)
(vii) \((x + 2)^3 = 2x(x^2 - 1)\)
(viii) \(x^3 - 4x^2 - x + 1 = (x - 2)^3\)
(i) Yes
(ii) Yes
(iii) No
(iv) Yes
(v) Yes
(vi) No
(vii) No
(viii) Yes
Quick reminder: A quadratic equation in \(x\) is any equation that can be written in the form \(ax^2 + bx + c = 0\) with \(a \neq 0\). That means, after expanding and bringing all terms to one side, the highest power of \(x\) must be 2.
(i) Check \((x + 1)^2 = 2(x - 3)\)
Step 1: Expand the left-hand side: \((x + 1)^2 = x^2 + 2x + 1\).
Step 2: Expand the right-hand side: \(2(x - 3) = 2x - 6\).
Step 3: Bring all terms to the left: \(x^2 + 2x + 1 - (2x - 6) = 0\), i.e. \(x^2 + 2x + 1 - 2x + 6 = 0\).
Step 4: Simplify: \(x^2 + 7 = 0\). Highest power is 2, so this is a quadratic.
(ii) Check \(x^2 - 2x = (-2)(3 - x)\)
Step 1: Expand the right-hand side: \((-2)(3 - x) = -6 + 2x\).
Step 2: Bring everything to the left: \(x^2 - 2x - (-6 + 2x) = 0\), i.e. \(x^2 - 2x + 6 - 2x = 0\).
Step 3: Combine like terms: \(x^2 - 4x + 6 = 0\). Highest power is 2, so this is a quadratic.
(iii) Check \((x - 2)(x + 1) = (x - 1)(x + 3)\)
Step 1: Expand LHS: \((x - 2)(x + 1) = x^2 - x - 2\).
Step 2: Expand RHS: \((x - 1)(x + 3) = x^2 + 2x - 3\).
Step 3: Move RHS to LHS: \(x^2 - x - 2 - (x^2 + 2x - 3) = 0\).
Step 4: Simplify: \(x^2 - x - 2 - x^2 - 2x + 3 = -3x + 1 = 0\). This is \(-3x + 1 = 0\), degree 1, so it is not quadratic.
(iv) Check \((x - 3)(2x + 1) = x(x + 5)\)
Step 1: Expand LHS: \((x - 3)(2x + 1) = 2x^2 - 5x - 3\).
Step 2: Expand RHS: \(x(x + 5) = x^2 + 5x\).
Step 3: Bring all terms to LHS: \(2x^2 - 5x - 3 - (x^2 + 5x) = 0\).
Step 4: Simplify: \(2x^2 - 5x - 3 - x^2 - 5x = x^2 - 10x - 3 = 0\). Highest power is 2, so this is a quadratic.
(v) Check \((2x - 1)(x - 3) = (x + 5)(x - 1)\)
Step 1: Expand LHS: \((2x - 1)(x - 3) = 2x^2 - 7x + 3\).
Step 2: Expand RHS: \((x + 5)(x - 1) = x^2 + 4x - 5\).
Step 3: Bring RHS to LHS: \(2x^2 - 7x + 3 - (x^2 + 4x - 5) = 0\).
Step 4: Simplify: \(2x^2 - 7x + 3 - x^2 - 4x + 5 = x^2 - 11x + 8 = 0\). Highest power is 2, so this is a quadratic.
(vi) Check \(x^2 + 3x + 1 = (x - 2)^2\)
Step 1: Expand RHS: \((x - 2)^2 = x^2 - 4x + 4\).
Step 2: Move RHS to LHS: \(x^2 + 3x + 1 - (x^2 - 4x + 4) = 0\).
Step 3: Simplify: \(x^2 + 3x + 1 - x^2 + 4x - 4 = 7x - 3 = 0\). Degree is 1, so this is not quadratic.
(vii) Check \((x + 2)^3 = 2x(x^2 - 1)\)
Step 1: Expand LHS: \((x + 2)^3 = x^3 + 6x^2 + 12x + 8\).
Step 2: Expand RHS: \(2x(x^2 - 1) = 2x^3 - 2x\).
Step 3: Bring RHS to LHS: \(x^3 + 6x^2 + 12x + 8 - (2x^3 - 2x) = 0\).
Step 4: Simplify: \(x^3 + 6x^2 + 12x + 8 - 2x^3 + 2x = -x^3 + 6x^2 + 14x + 8 = 0\). Degree is 3, so this is not quadratic.
(viii) Check \(x^3 - 4x^2 - x + 1 = (x - 2)^3\)
Step 1: Expand RHS: \((x - 2)^3 = x^3 - 6x^2 + 12x - 8\).
Step 2: Move RHS to LHS: \(x^3 - 4x^2 - x + 1 - (x^3 - 6x^2 + 12x - 8) = 0\).
Step 3: Simplify carefully: \(x^3 - 4x^2 - x + 1 - x^3 + 6x^2 - 12x + 8 = 2x^2 - 13x + 9 = 0\).
Step 4: Highest power of \(x\) is 2, so this is a quadratic.
Bottom line: For each equation: (1) expand both sides, (2) move all terms to one side, (3) simplify, and (4) look at the highest power of \(x\). If it is 2 and the \(x^2\) coefficient is not zero, the equation is quadratic.