5Number 5ExplanationThe roots of the equation x2+ax−4=0x^2 + ax - 4 = 0x2+ax−4=0 are ppp and qqq If p2−2pq+q2=8ap^2 - 2pq + q^2 = 8ap2−2pq+q2=8a, the value of aaa that satisfies is ....555444222111−2-2−2ExplanationGiven the quadratic equation x2+ax−4=0x^2 + ax - 4 = 0x2+ax−4=0 with roots ppp and qqq Based on Vieta's formulas, we know that p+q=−ba=−ap + q = -\frac{b}{a} = -ap+q=−ab=−ap⋅q=ca=−4p \cdot q = \frac{c}{a} = -4p⋅q=ac=−4 If p2−2pq+q2=8ap^2 - 2pq + q^2 = 8ap2−2pq+q2=8a, then we can simplify the left side p2+q2−2pq=8ap^2 + q^2 - 2pq = 8ap2+q2−2pq=8a(p+q)2−2pq−2pq=8a(p + q)^2 - 2pq - 2pq = 8a(p+q)2−2pq−2pq=8a(p+q)2−4pq=8a(p + q)^2 - 4pq = 8a(p+q)2−4pq=8a Substitute the values p+q=−ap + q = -ap+q=−a and pq=−4pq = -4pq=−4 (−a)2−4(−4)=8a(-a)^2 - 4(-4) = 8a(−a)2−4(−4)=8aa2+16=8aa^2 + 16 = 8aa2+16=8aa2−8a+16=0a^2 - 8a + 16 = 0a2−8a+16=0 Factor the quadratic equation (a−4)2=0(a - 4)^2 = 0(a−4)2=0a=4a = 4a=4 Therefore, the value of aaa that satisfies is 444
