0000:00:00:00:00Start15Number 15ExplanationAn equation is expressed in the form ∣x2x21∣=8\begin{vmatrix} x^2 & x \\ 2 & 1 \end{vmatrix} = 8x22x1=8. The possible values of xxx are...4 or −24 \text{ or } -24 or −2−4 or 2-4 \text{ or } 2−4 or 2−2 or 3-2 \text{ or } 3−2 or 32 or −32 \text{ or } -32 or −33 or 83 \text{ or } 83 or 8ExplanationGiven the matrix determinant equation ∣x2x21∣=8\begin{vmatrix} x^2 & x \\ 2 & 1 \end{vmatrix} = 8x22x1=8. Recall that the determinant of a 2×22 \times 22×2 matrix is the difference between the product of the main diagonal and the secondary diagonal: ∣abcd∣=ad−bc\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bcacbd=ad−bc. So, we expand the equation: (x2)(1)−(x)(2)=8(x^2)(1) - (x)(2) = 8(x2)(1)−(x)(2)=8x2−2x=8x^2 - 2x = 8x2−2x=8 Move all terms to the left side to form a quadratic equation: x2−2x−8=0x^2 - 2x - 8 = 0x2−2x−8=0 Factor the quadratic equation. We look for two numbers that multiply to −8-8−8 and add up to −2-2−2. These numbers are −4-4−4 and 222. (x−4)(x+2)=0(x - 4)(x + 2) = 0(x−4)(x+2)=0 The zeros are: x−4=0⇒x=4x - 4 = 0 \Rightarrow x = 4x−4=0⇒x=4x+2=0⇒x=−2x + 2 = 0 \Rightarrow x = -2x+2=0⇒x=−2 So, the possible values of xxx are 444 or −2-2−2.
15Number 15ExplanationAn equation is expressed in the form ∣x2x21∣=8\begin{vmatrix} x^2 & x \\ 2 & 1 \end{vmatrix} = 8x22x1=8. The possible values of xxx are...4 or −24 \text{ or } -24 or −2−4 or 2-4 \text{ or } 2−4 or 2−2 or 3-2 \text{ or } 3−2 or 32 or −32 \text{ or } -32 or −33 or 83 \text{ or } 83 or 8ExplanationGiven the matrix determinant equation ∣x2x21∣=8\begin{vmatrix} x^2 & x \\ 2 & 1 \end{vmatrix} = 8x22x1=8. Recall that the determinant of a 2×22 \times 22×2 matrix is the difference between the product of the main diagonal and the secondary diagonal: ∣abcd∣=ad−bc\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bcacbd=ad−bc. So, we expand the equation: (x2)(1)−(x)(2)=8(x^2)(1) - (x)(2) = 8(x2)(1)−(x)(2)=8x2−2x=8x^2 - 2x = 8x2−2x=8 Move all terms to the left side to form a quadratic equation: x2−2x−8=0x^2 - 2x - 8 = 0x2−2x−8=0 Factor the quadratic equation. We look for two numbers that multiply to −8-8−8 and add up to −2-2−2. These numbers are −4-4−4 and 222. (x−4)(x+2)=0(x - 4)(x + 2) = 0(x−4)(x+2)=0 The zeros are: x−4=0⇒x=4x - 4 = 0 \Rightarrow x = 4x−4=0⇒x=4x+2=0⇒x=−2x + 2 = 0 \Rightarrow x = -2x+2=0⇒x=−2 So, the possible values of xxx are 444 or −2-2−2.
